
b£>3 



GopyrightN°_ 



COPYRIGHT DEPOSIT. 



MANUAL 

OF 



Reenforced 
Concrete Construction 

BY 

EDMOND G. F. R_. DU MAZUEL, B.L., D.Sc, L.Sc. 



Formerly Constructing Engineer, Roads of Isere Section, 
France ; Chief of Staff, Military Improvements of the 
Harbor of Algiers, Algeria ; Non-Resident Lecturer, 
Ecole Polytechnique. Consulting Engineer, New York 
City, in charge during construction, of important works in 
America such as American Asbestos Mills, at Black Lake, 
Canada ; Power Stations for Iroquois Construction Co., at 
Niagara Falls ; Power Stations and Transmission Lines 
for Niagara, Lockport & Ontario Power Co., etc., etc. 

Second Edition, Fifteenth Thousand 
PRICE ONE DOLLAR 



COPYRIGH t 19 10 

B y E . G . F . R — du Mazuel 

ALL RIGHTS ASSIGNED TO THE 

CONCRETE INTERLOCKING STEEL CORPORATION 

NEW YORK CITY 

Manufacturers of the du Mazuel Reenforcements for Concrete 






^ 



/0-3JOI3 

CCI. A 27831 9 



PREFACE 

Concrete is rapidly replacing other materials for 
building purposes. 

This publication as a whole, more thoroughly than 
any previous one of its kind, sets forth briefly the 
principles and methods of concrete architecture. 

Reenforced concrete construction is a compara- 
tively new subject in America. An engineer, archi- 
tect or builder, therefore, need not as yet feel 
ashamed of lacking a working knowledge of this 
method of building, which is so fast superceding 
other forms. 

The Manual is intended to be of practical assist- 
ance to engineers, architects and builders in design- 
ing and constructing reenforced concrete. Its pur- 
pose is to give facts, not theories, and be so clear 
and concise that even a person unaccustomed to 
building may understand. 

It should not be lost sight of that there must be 
in any kind of construction, constant and competent 
inspection of the work. In reenforced concrete 
work this inspection is not so essential as in non- 
reenforced concrete, but it is still necessary. 



CONCRETE 
PLAIN AND REENFORCED 

Going over the various materials that are known 
to have been used from almost the time when men 
abandoned caves and took to building their homes 
in the open, it is interesting to hnd that con- 
crete, in a rough state, was the material used even be- 
fore taking to log, frame, stone, and brick buildings. 
The value of concrete as a substitute for timber, 
stone, brick and iron foundations and superstruc- 
tures is now well known, and the situation will be 
found exceedingly rare where this substance can- 
not be successfully employed, because it becomes 
petrified and thereby a one-stone structure. The 
pyramids of Egypt, particularly, are old and well 
known, and may be cited as an example of excel- 
lent concrete work, and as having been fairly tried 
by the test of time. Pieces of the giant blocks 
which have occasionally been taken by scientists, 
and by the writer himself, have by test shown them- 
selves to be not inferior in hardness to some of the 
serpentine rocks of North America. A number of 
these tested pieces from the pyramids contained 
nummulites, which fossil shells were the petrified 
remains of the lentils which had been used as food 
by the workmen employed in the construction of 
these vast edifices and which had been thrown into 
the concrete mixers in accordance with the rites of 
their religion. 



Naturally, the kind of cement used is of vital im- 
portance and is a matter that should always be 
considered with great care. It is to be regretted 
that some of the old and valuable formulae for ce- 
ment used by the Mongolians, Egyptians and Ro- 
mans were destroyed in the great fires of Alexan- 
dria and other old cities. However, science is giv- 
ing us from day to day a better and better cement. 

Where, in the course of time, iron and steel will 
rust and weaken materially, timber rot, and bricks 
flake and pulverize, concrete, when properly made, 
will stand for ages, being practically stone — for af- 
ter all, stone is concrete petrified by nature. 

It has been found that a very great degree of 
added strength can be secured by embedding iron or 
steel of various shapes in concrete. Two very great 
advantages of thus reenforcing concrete are : firstly, 
that the weight and size of a given structural mem- 
ber need be much less when reenforced than when 
plain; and, secondly, member for member, greater 
uniformity of strength is secured, making a safer and 
more reliable construction. 

There are now in the market scores of different 
forms of reenforcement, such as bars — twisted, 
lugged, dented, etc. ; steel sheets — bent, twisted, ex- 
panded, etc. ; and wire cloth criss-crossed in every 
conceivable fashion. The author, in his professional 
practice as a general consulting engineer, found that 
every known form of reenforcement had grave dis- 
advantages, the main ones being that their shape pre- 
vented their giving the distribution of strength 
where it belonged ; that they required the use of ex- 
pensive preliminary forms, the same as in plain con- 



crete, the forms being of little or no value after once 
having been used, and that for a given degree of 
strength they required too much dead weight of 
materials. 

To get a reenforcement that overcame the disad- 
vantages just enumerated, the author, after much in- 
vestigation, designed two kinds of reenforcements ; 
one, the du Mazuel plain reenforcement made from 
flat bessemer or open hearth process steel into the 
peculiar reversed and double "S" form which brings 
the steel area where it belongs as a reenforcement, 
and the other the du Mazuel expanded reenforce- 
ment, made from any of the several kinds of ex- 
panded metal on the market, corrugated in such 
manner as to give it the greatest strength with the 
least amount of material. 



CONCRETE MIXTURES 



In any concrete structure, no matter how reen- 
forced, the efficiency of the finished work depends 
largely on the selection of the materials, the propor- 
tions in which they are used and the care with which 
the work is executed. 

Long experience has shown that best results have 
been had with mixtures of one cement to three or 
four of crushed stone; the crushed stone to be sound 
and to pass one-quarter inch mesh. These mixtures 
in thin floors and roofs have been found by actual 
tests to give fhe most satisfactory results with du 
Mazuel reenforcements, a one to three stone con- 
crete having been used in arriving at the data of the 
diagrams or curves given herein. Where good 
crushed stone (trap rock or granite) is not avail- 
able, a well graded sand or gravel will answer the 
purpose very well. 

If in the judgment of the engineer, architect or 
designer a less rich grade of concrete is preferable, 
it must be remembered that the stone (or cinders if 
so desired) in the concrete filling the corrugations, 
must be of such size as to readily enter the con- 
tracted area between the corrugations, as otherwise 
voids would occur which would be detrimental to 
the strength of the structure. 

Within recent years the writer has successfully 
used a waste product as a substitute for sand and 
stone, namely, iron ore tailings. It has been found 



as a rule to be a well graded material that produced 
a very satisfactory mortar or concrete. 

When it is desired to color the concrete so as to 
produce certain effects, this may be done by thor- 
oughly mixing Avith the cement, before adding the 
sand or stone, pulverized iron oxide of the basic 
color wanted and proportioning to the blend or tint 
required. 

Tables of concrete mixtures and proportions, 
given herein, will be found to be of assistance in 
actual office and field practice. 



PROPORTIONS FOR ONE CU. YD. OF CONCRETE 


MIXTURES 


STONE 2^" 

SCREENED 


GRAVEL Y±" AND 
UNDER 


Cement 


Sand 


Stone 


Cement 
Bbls. 


Sand 
Cu.Yds. 


Stone 
Cu.Yds. 


Cement 
Bbls. 


Sand 
Cu. Yds. 


Stone 
Cu. Yds. 








































2 


3 


175 


.52 


.80 


1.55 


.47 


.72 




2 


4 


1.53 


.46 


.93 


1.35 


.42 


.84 




3 


5 


1.25 


.52 


.92 


1.13 


.48 


.83 




3 


6 


1.11 


.45 


1.00 


LOO 


.40 


.90 




4 


6 


1.00 


.53 


.95 


.92 


.48 


.87 




4 


8 


.82 


.50 


1.02 


.75 


.43 


.92 



10 



PROPORTIONS FOR ONE CU. YD. MORTAR 


MIXTURES 


WET UN- 
RAMMED 


RAMMED 


Cement 


Sand 


Cement 
Bbls. 


Sand 
Cu.Yds. 


Cement 
Bbls. 


Sand 
Cu.Yds. 




1. 


4.7 


.7 


5.4 


.8 




1.5 


3.7 


.8 


4.2 


1.0 


1 


2. 


3.0 


.9 


3.4 


1.0 




2.5 


2.6 


1.0 


2.9 


1.1 




3. 


2.2 


1.0 


2.5 


1.1 




3.5 


1.9 


1.0 


2.2 


1.2 




4. 


1.7 


1.1 


2.0 


1.2 




4.5 1.4 


1.1 


1.9 


1.3 



In many instances of special construction the ma- 
jority of engineers and architects are strongly 
averse to the use of concrete, and to change this feel- 
ing the author has worked hard and with marked 
success for the last few years. 



For silos, cisterns, live stock drinking troughs or 
feeding floors, liquid manure wells or sinks, sewers 
or conduits, where putrefaction or disease germs may 
generate, accumulate and flourish ; and where a de- 
sign is for terrazzo floors or platforms and special 
roofs ; the author has used a concrete which in some 
of the warm countries of Europe has been called the 
du Mazuel oleaginous concrete. 

This concrete is the ordinary kind to which, im- 
mediately after being mixed, non-volatile mineral 
oil is added in the proportion of 12 per cent, of oil 
by weight, to the weight of the cement, the mass be- 
ing kept agitated for a few minutes after oil is added. 

This concrete is very effective, as it is perfectly 
waterproof, shows no cracks known as "worm 
cracks", and gives a dustless, waterproof and in 
many instances acidproof surface. White oil may 
be substituted for non-volatile mineral oil when the 
latter cannot be procured. The oil slightly delays 
the initial and final setting. It also reduces the ten- 
sile strength in about the proportion which the 
amount of oil bears to the whole mass. The oil, al- 
kalies and water form an emulsion which easily 
combine in the concrete. The mixing of non-vola- 
tile mineral oil with concrete is exceedingly simple. 



WATER 

Too little care is given to the water used by the 
average contractor or builder when making con- 
crete. The author believes that a small percentage 
of the failures in concrete is due to that fact. 



Only pure, clear water should be used in cements, 
mortars or concrete. 

When the temperature is at 32 degrees Fahren- 
heit the contractor or builder should dissolve two 
pounds of pulverized calcium chloride (CaCl 2 ) in the 
water necessary for a batch, utilizing one barrel of 
hydraulic Portland cement; and for each degree un- 
der 32, the amount should be increased by two 
ounces of pulverized calcium chloride (CaCL). 

The only effect that calcium chloride (CaCl 2 ) has 
on concrete, apart from preventing freezing, is to 
slightly retard the initial set, and at the same time to 
slightly increase its strength. 



COMPARATIVE 
CONSTRUCTION COSTS 

As the element of cost is always one of prime im- 
portance, the following table was carefully compiled 
to show at a glance the cost per cubic foot of various 
kinds of buildings actually constructed since 1905. 
In making estimates of cost all dimensions of the 
proposed building should be taken from extreme out- 
side lines. 



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DuMAZUEL REENFORCEMENTS 



Du Mazuel reenforcements commend themselves 
by their great strength, light weight, perfect self- 
interlocking, low cost and adaptability to all construc- 
tions where concrete is to be used. 

The great strength and light weight of these reen- 
forcements are clearly shown when compared with 
others, as the lightest of constructions with other 
reenforcement capable of sustaining the same live 
loads, is more than five times the dead weight of 
the du Mazuel construction. 

The low cost is also readily seen, as these reen- 
forcements eliminate all form work, all skilled labor, 
reduce the whole dead weight of a building, and 
the plain reenforcement, even before any concrete 
is applied, forms a water-tight structure, a matter 
of great moment to contractors, who never have 
enough shelter for their materials. 

The uses to which these reenforcements may be 
put are innumerable and here will be mentioned only 
a few of the places where their value has been de- 
monstrated. A great many other uses for these ma- 
terials will suggest themselves to the engineer, archi- 
tect or designer. 

All of the lighter gauges are especially adapted 
for reenforcing floors, roofs, walls, partitions, ceil- 
ings, buttresses, abutments, retaining walls, dams, 
penstocks, culverts, flues, chimneys, fire places, boil- 
er settings, coal bins, hoppers, sidewalk slabs, etc. — 
even for reenforcing concrete canal boats and bargee 

The heavier gauges are used to advantage for 
bridge work, sheet piling, coffer dams, tunnel 
tubing, trench and pit lining or sheeting, etc. These 
heavier gauges make light, stiff and inexpensive 



sheet pile, which requires no special tools, clips, 
joints, or shop work. 

Plain du Mazuel reenforcement is furnished in 
United States standard gauges from number 18 to 
28, inclusive; approximately one foot wide and up to 
twelve feet in length, the heavier gauges such as 18 
and 20 being used for sheet piling. 

The common radius of reversed "S" curves in 
du Mazuel reenforcement can be easily computed as 
follows : 

g = width of section or half pitch. 

r = common radius. 

k = depth of reenforcement. 

0.25 (g 2 +k 2 ) 

r = 

k 

Expanded du Mazuel reenforcement is furnished 
in United States standard gauges from number 20 
to 28; twelve inches and wider and eight feet in 
length. 

Inquiries as to special requirements in either kind 
of reenforcement, as to gauge, width and length are 
given careful attention by the manufacturers. 

All of the du Mazuel reenforcements are protected 
by patents in the United States and foreign countries. 



USE WITHOUT FORMS 

One of the greatest advantages possessed by the 
du Mazuel reenforcements is the ability to do with- 
out wooden or other forms. In tunnel and pipe 
work, these reenforcements may be used as an in- 
side and outside form. If thick walls are to be con- 
structed, it may be found economical to use the re- 
enforcements simply as walls of a form, plastering 
the outsides and filling in between with concrete 
mortar. 



18 




View of a slab reenforced with Standard du Mazuel 
Reenforcement made of Plain Sheets 



(Standard No. 22, or No. 24. or No. 26, or No. 28.) 



19 




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View of slab reenforced with Standard du Mazuel Reenforcement 
made of Expanded Metal 



(Standard No. Lxl8, or No. Lx20, or No. £x22, or No. tx24 or No. Lx26.) 




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INTERLOCKING 

In joining, overlapping or interlocking the du 
Mazuel reenforcements on the ends, the most con- 
venient way will be found to press the reenforce- 
ment sheets into position with the foot, or with a 
mallet and wooden block. By placing one's heel 
against the outside corrugations of the top sheet, 
they will readily sink into place and start the inter- 
locking of the adjacent corrugations. Some difficulty 
may be experienced if an attempt is made to inter- 
lock the entire width of the sheets at once, but by 
pressing the corrugations down consecutively no 
trouble will be found. When joining the sheets 
lengthwise, as the outside corrugation only is lap- 
ped, there is no difficulty in interlocking the entire 
length of the sheet at the same time. 



23 





THE DU MAZUEL RE-ENFORCEMENT CAN BE LAID ON TOP 
CORDS OF GIRDERS, WHERE THEY MAY BE SECURED BY TURN- 
ING BACK OVER FLANGE "TONGUES" CUT OUT FROM RE- 
ENFORCEMENT WITH A SHARP V-SHAPED CHISEL. 

A SIMILAR BINDING CAN BE MADE TO FURTHER SECURE 
TWO RE-ENFORCEMENT SHEETS WHEN SAME ARE OVER- 
LAPPED BETWEEN SPANS. 



FASTENING TO SUPPORTING STRUCTURE 

For floors and flat roofs, it will not be found neces- 
sary to fasten the du Mazuel reenforcement sheets 
to the beams, as the interlocking of the sheets has 
the effect of making them similar to one continuous 
sheet over the whole surface. The weight of the 
concrete on top and the plastering on the bottom 
and around the beams, effectively prevents any pos- 
sibility of the floor sliding on the top cords of the 
beams. For pitch roofs, walls, partitions, etc., the 
sheets may be nailed or bolted to a furring strip on 
the purlins, columns or the like, or fastened as is 
ordinary corrugated iron. In cases of sewers, retain- 
ing walls, dams, etc., the materials may often be used 
as a portion of the original form as well as a 
reenforcement. 



IS A FLOOR BEFORE CONCRETING 

The du Mazuel reenforcements when used in roof 
and floor work, themselves act as a floor, on which the 
necessary work of depositing the concrete may 
be carried on without having other support. 
Unless long spans are used the du Mazuel 
reenforcements are amply strong to carry the load 
of workmen, wheelbarrows and materials. It is 
well, however, to use boards or planks for runways 
when wheelbarrows are used. For very long spans 
and the lighter gauges of material it will be found 
better to support the sheets as necessary with a 
light frame work at centre until the top coating of 
concrete has set. 



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PLASTERING 

The du Mazuel reenforcements are furnished 
in plain or expanded metal sheets, either black, 
painted or unpainted, or galvanized. If black sheets 
are chosen they should receive the same care as in 
the handling of any other black sheet product. Sheet 
steel will corrode if left unpainted or unprotected; 
and where the material is likely to lay around for 
a considerable length of time before it is used, it is 
advisable to have the sheets painted, which can be 
done at a very small additional cost. 

It is not best to leave the under side of the du 
Mazuel reenforcements unplastered, even in cases 
where the concrete on one side is sufficient to carry 
the loads required. No better protection for steel 
has ever been devised than a good coating of con- 
crete. This is easily understood, as concrete made 
out of Portland or hydraulic cement maintains with- 
in itself a uniform alkaline reaction providing the 
free lime has not been removed by percolating 
waters, etc. ; that is to say, as long as it remains a 
concrete body. To this end all porous materials, 
such as cinders, etc., should be avoided where the 
concrete is exposed to the elements. 

In the case of a structure which is not intended to 
be permanent, the under side of the materials may 
be painted or galvanized, as either of these methods, 
if properly done, will enable them to withstand cor- 
rosion for a long time. This, of course, should be 
done only when the concrete on one side is sufficient 
to carry the load. 

In plastering the under surface, care should be 
exercised to completely fill the corrugations, as in 
order to develop the full strength of the du Mazuel 
slabs, compression must occur along the diagonal 
lines connecting the centres of the circular parts of 



the corrugations. This compression in what is 
ordinarily the tension zone of a reenforced slab, is a 
peculiar feature of this reenforcement. The du 
Mazuel slabs cannot collapse unless the corrugations 
were to burst, or the concrete crush. 

The plastering which is below the bottom of the 
corrugations is not to be considered, its purpose 
being merely to protect the metal against corrosion 
or fire. The depth to which this is put on is a mat- 
ter of individual judgment. In the official fire and 
load tests on the du Mazuel reenforcements, the 
plastering extended three-eighths of an inch below 
the bottom of the corrugations. 

In plastering, a one to two concrete mixture is 
recommended, with the addition of a small amount 
of metallurgical hair. Burnt lime may also be added 
in the proportion of one-tenth of the volume of ce- 
ment used, but if the work is subject to hazardous 
fire conditions, it is better to use only the hair on ac- 
count of the danger of dehydration of the lime, 
which will crumble upon losing its molecular 
moisture. 



ROOFS — WATERPROOFING 

As a means of waterproofing as well as reenforc- 
ing a concrete structure, the. du Mazuel reenforce- 
ments offer as perfect a system as has ever been de- 
vised. The interlocking features of the du Mazuel 
reenforcement made of plain sheets, make the joints 
practically watertight before even the concrete is 
applied. Engineers have found that for pitch roofs 
it is not necessary, if the work is properly done, 
to finish the surface with any sort of prepared roof- 
ing, felt or other waterproofing. These du Mazuel 
sheets without concrete offer a far more perfect cor- 
(see page 33) 



31 



(U UJ 



„0 .06 -IBlSUIEia-^ 





Re-enforcement fastened 
to flange of tee. 





w 

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A \\\ 


Tee Rings to be jl h 


8' or 


12' centers 1 




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i HI 




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du Mazuel 
re-enforcement ->. 


v ft 

A/' 

w / 

/ 

/ 

/ 
/ 








CONCRETE PIPE RE-ENFORCED AND INSIDE 
FORMS DONE AWAY WITH BY THE USE OF 
DU MAZUEL RE-ENFORCEMENT. 



33 

rugated roofing than the ordinary corrugated iron. 
The addition of the concrete to the top surface and 
the plaster to the bottom, preserves the sheets 
against any possibility of corrosion and makes a 
permanent and indestructible, acidproof, fireproof 
and waterproof roof. 

For flat roofs where water may accumulate it is 
desirable to cover the top coating of concrete with 
roof felt or prepared roofing or to finish with tar and 
gravel. The author on several occasions has water- 
proofed roofs by saturating every part of the surface, 
after it thoroughly dried, with hot paraffin wax, 
using hot irons to press in the wax. As the wax 
works its way in it will stop up all of the pores of the 
concrete, and form a perfect nonleakable surface that 
cannot be injured by frost or sun, and this at a very 
nominal cost. 

A few suggestions as to roof details are shown 
on various plates herein. For such places as stack 
and ventilator openings, skylights, etc., the designer 
will find no more difficulty in flashing and water- 
proofing them than with any other form of roof. 



PARTITIONS 

When du Mazuel expanded reenforcement is used 
for partition work, there is not only a great saving 
in the cost of the work, but also a great saving of 
time, as with this material the body of the wall is 
secured with the first coat of plaster. 

In using either kind of reenforcement for side 
walls or partitions they are preferably placed with 
the corrugations running horizontally, as they then 
present a more easily plastered surface. 



-Wood or steel runner 




INTERIOR OF A ROOM SHOWING USES OF 
DU MAZUEL RE-ENFORCEMENT IN RESIDENCES, ETC.; TWC , DETAILS FOR A 

AND, TYPICAL DETAIL OF FLOOR THAT IS NOT 
REQUIRED TO CARRY HEAVY LOADING. 



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-du Mazuel re-enforcement 



PLAN SHOWING WALL WITH DU MAZUEL 
RE-ENFORCEMENT USED AS FURRING 



THICKNESS OF WALLS IN INCHES, FOR MERCANTILE 
BUILDINGS AND LIVERY STABLES, AND, EXCEPT IN CHICAGO, 
FOR ALL BUILDINGS OVER FIVE STORIES IN HEIGHT. 



Height of 
Building 



Two 

Stories 



Three 

Stories 



Four 

Stories. 



Five 

Stories 



Six 
Stories 



Seven 
Stories 



Eight 
Stories 



f Boston 

I New York 

I Chicago. ...... 

J Minneapolis. . . 
[ St. Louis 

Denver 

i San Francisco. , 
I. New Orleans. . . 
'Boston 

New York 

Chicago 

Minneapolis. . . 

St. Louis 

Denver 

San Francisco. 

New Orleans. . 

Boston 

New York 

Chicago 

Minneapolis. . . 

St. Louis 

Denver 

San Francisco. 

New Orleans. . 
f Boston 

New York 

Chicago. ...... 

J Minneapolis. . . 
} St. Louis 



Denver. 
I San Francisco. 
(.New Orleans. . , 
'Boston 

New York 

Chicago 

Minneapolis. . . 

St. Louis 

Denver 

San Francisco. 

New Orleans. . 

Boston 

New York. .... 

Chicago 

Minneapolis. . . 

St. Louis.. t ,\ . 

Denver 

New Orleans. . , 

fBoston 

j New York .... 

I Chicago 

•\ Minneapolis. . . 

St. Louis. ..... 

I Denver. ...... 

I New Orleans. . , 









Stories 








1st. 


2d. 


3d. 


4th. 


5th. 


6th. 


7th. 


8th. 


16 


12 














12 


12 














12 


12 














12 


\1 














IS 


13 














13 


13 














17 


13 














13 


13 














20 


16 


16 












16 


16 


12 












16 


12 


12 












16 


12 


12 












18 


18 


13 












17 


17 


13 












17 


17 


13 












13 


13 


13 












20 


16 


16 


16 










16 


16 


16 


12 










20 


16 


16 


12 










16 


16 


12 


12 










22 


18 


18 


13 










21 


17 


17 


13 










17 


17 


17 


13 










18 


18 


13 


13 










20 


20 


20 


20 


16 








20 


16 


16 


16 


16 








20 


20 


16 


16 


16 








20 


16 


16 


12 


12 








22 


22 


18 


18 


13 








21 


21 


17 


17 


13 








21 


17 


17 


17 


13 








18 


18 


18 


13 


13 








24 


20 


20 


20 


20 


16 






24 


20 


20 


20 


16 


16 






20 


20 


20 


16 


16 


16 






20 


20 


16 


16 


16 


12 






26 


22 


22 


18 


18 


13 






26 


21 


21 


17 


17 


13 






21 


21 


17 


17 


17 


13 






22 


18 


18 


18 


13 


13 






24 


20 


20 


20 


20 


20 


16 




28 


24 


21 


20 


20 


16 


16 




20 


20 


20 


20 


16 


16 


16 




20 


20 


20 


16 


16 


16 


12 




26 


26 


22 


22 


18 


18 


13 




26 


21 


21 


21 


17 


17 


17 




22 


22 


18 


18 


18 


13 


13 




28 


24 


20 


20 


20 


20 


20 


16 


32 


28 


24 


24 


20 


20 


16 


16 


24 


24 


20 


20 


20 


16 


16 


16 


24 


20 


20 


20 


16 


16 


16 


12 


30 


26 


26 


22 


22 


IS 


18 


13 


30 


26 


21 


21 


21 


17 


17 


17 


22 


£2 


22 


18 


18 


18 


13 


13 



39 



Height of 
Building 



Stories 


j 






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20 


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20 


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32 


32 


28 


24 


24 


20 


20 


10 


10 








24 


24 


24 


20 


20 


20 


10 


10 


10 








24 


24 


20 


20 


20 


10 


10 


10 


12 








m 


30 


20 


20 


22 


22 


18 


18 


13 








30 


20 


20 


21 


21 


21 


17 


17 


17 








28 


28 


24 


24 


20 


20 


20 


20 


20 


10 






30 


32 


32 


28 


2-1 


24 


20 


20 


10 


10 






2x 


2.S 


24 


24 


2; 


20 


20 


20 


10 


10 






_!- 


24 


24 


20 


20 


20 


10 


10 


10 


12 






3 J 


30 


30 


20 


20 


22 


22 


IS 


18 


13 






30 


30 


20 


20 


21 


21 


21 


17 


17 


17 






36 


32 


32 


28 


28 


24 


20 


20 


20 


20 


10 




36 


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32 


2S 


28 


24 


24 


20 


20 


10 


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34 


34 


30 


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22 


18 


18 


13 




30 


30 


20 


20 


20 


21 


21 


21 


17 


17 


17 




30 


30 


32 


32 


28 


28 


24 


20 


20 


20 


20 


16 


40 


33 


30 


32 


32 


"8 


24 


24 


20 


20 


10 


10 


2S 


20 


OS 


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24 


24 


20 


20 


20 


10 


1(5 


10 


N-: 


34 


34 


30 


30 


•:•:■■ 


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22 


2 V 


18 


18 


13 


30 


30 


30 


20 


20 


20 


21 


21 


21 


17 


17 


17 



f Boston. . . . 
New York. . 
Nine j Chicago. . . . 
Stories | Minneapolis 
I St. Louis. .. 
I Denver. . . . 

f Boston 

I New York. . 



Ten 

Stories 



Eleven 
Stories 



Twelve 
Stories 



i 



Chicago. 

Minneapolis. 

St. Louis. .. . 

(, Denver 



Boston. . . . 
New York. . 
Chicago. . . . 
St. Louis. . . 
Denver. . . . 



("Boston. . . . 
New York.. 



(Chicago. 
St. Louis. 
Denver. . . 



THICKNESS OF ENCLOSING WALLS, FOR 
RESIDENCES, TENEMENTS, HOTELS, AND OFFICE 
BUILDINGS —CHICAGO BUILDING ORDINANCE 





1 


Stories 


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Basement and 


12 
12 
16 
20 
20 
20 
24 
24 
28 
28 
28 
32 


8 
12 
12 
16 
16 
20 
24 
24 
24 
24 
2S 
28 


S 
12 
16 
16 
16 
20 
24 
24 
24 
24 
28 


8 
12 
16 
10 
20 
20 
20 
24 
21 
24 


12 
12 
16 
16 
20 
20 
20 
24 
24 


12 

12 
16 
16 

20 
20 
20 
24 


12 
12 
16 

16 
20 
20 
20 


12 
12 
10 
10 
20 
20 


12 

12 
13 
10 
20 


12 
12 
10 
10 


12 
12 
16 


12 
12 


Three-story 

Four-story. 

Five-story 


Seven-story 

. Eight -story 

Nine-story , 


Eleven-story , 

Twelve-story 



NOTES 
ON THE DESIGN OF FLOORS 

The following are the usual assumptions made in 
good practice for superimposed loads on floors. 



STANDARD LOADING 

Dwellings and offices; 70 pounds per square foot. 

Churches, theatres and ball rooms; 125 pounds 
per square foot. 

Warehouses and factories ; 200 to 250 pounds per 
square foot. 

Heavy machinery; not less than 250 pounds per 
square foot. 

It will usually be found that the building laws of 
various cities conform to the above. 



FLOOR FINISH 

For determining the dead loads of the different 
kinds of floor finish, the following weights per 
square foot may be used : 

Spruce, y% ; 2.1 pounds per square foot. 

Southern pine or maple, %" ; 4 pounds per square 
foot. 

Sleepers, 3" x 4", 16" centers ; 2.5 pounds per square 
foot. 



Marble, i" thick; 14 pounds per square foot. 
Cinder fill, per inch ; 4 pounds per square foot. 
Cinder concrete, per inch ; 8 pounds per square foot. 
Suspended ceilings; 10 to 20 pounds per square 
foot. 

Plaster, %" ; 2 to 4 pounds per square foot. 



SPACING OF STRUCTURAL MEMBERS 

The du Mazuel reenforcements being furnished 
in sheets of any length up to 8 feet in the expanded 
and 12 feet in the plain form, the designer is able 
to space the columns, uprights, girders, etc., in the 
most economical manner. The du Mazuel sheets 
are laid directly against the structural members, 
which may be of any standard section and fastened 
as previously described, no furring or intermediate 
supports being required. In plastering the sides or 
undersides care should be taken to work the mortar 
well into the corrugations along and against the 
structural members. 



SAFE LIVE LOADS, 
DEAD LOADS, WEIGHTS, ETC. 

The diagrams or curves given in this Manual 
are constructed upon well known formulae, using 
coefficients that are universally accepted as safe 
for conditions to which they are intended to apply, 
and furthermore are primarily made from actual 
tests. 

These diagrams or curves, giving not only the 
weight of the reenforcements, the dead weight of 
the various slabs, but also the safe live loads per 
square foot for given spans, will enable the engineer, 
architect or designer to quickly and accurately fill 
his requirements without the usual lengthy and 
tedious calculations. 

In all general cases it will be found that a con- 
siderable saving on the total cost of construction, 
as well as a great saving on the floors and slabs 
themselves, can be effected in the design of the steel 
supporting structure, due to the reduction of the 
dead load of the floors and slabs, when these are 
designed to use the du Mazuel system. 



43 



CURVES FOR SLABS RE-ENFORCED 

WITH STANDARD No. 22 DU HAZUEL RE-ENFORCEMENT 

WEIGHING 3.84 POUNDS PER SQUARE FOOT. 



* * * is 









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NOTE— For slabs re-enforced with standard No. Exl8du Mazuel re-enforcement -take two-thirds 
of given loads as safe live loads. 



o. ft 
•3 -o 



CURVES FOR SLABS RE-ENFORCED 

WITH STANDARD No. 24 DU riAZUEL RE-ENFORCEMENT 

WEIGHING 3.07 POUNDS PER SQUARE FOOT. 



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NOTE— For slabs re-enforced with standard No. Ex20 du Mazuel re-enforcement take two-thirds 
of given loads as safe live loads. 



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CURVES FOR SLABS RE-ENFORCED 
2 £ WITH STANDARD No. 26 DU HAZUEL RE-ENFORCEMENT 

c c WEIGHING 2.30 POUNDS PER SQUARE FOOT. 



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NOTE— For- slabs re-enforced with standard No. Ex22du Mazuel re-enforcement take two-thirds 
of given loads as safe live loads. 



46 



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CURVES FOR SLABS RE-ENFORCED 

WITH STANDARD No. 28 DU rtAZUEL RE-ENFORCEflENT 

WEIGHING 1.92 POUNDS PER SQUARE FOOT. 



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NOTE— For slabs re-enforced with standard No. Ex24 du Mazuel re-enforcement take two-thirds 
of given loads as safe live loads. 



47 
NOTE 

It must be borne in mind if du Mazuel Reenforce- 
ment made of expanded metal is selected for 
reenforcing floor and roof slabs, that these slabs do not 
reach their full bearing capacity before thirty to ninety 
days have elapsed after initial set of the concrete. 

All of the curves refer to du Mazuel slabs of I to 3 
stone concrete. L represents the clear span in feet; 
a, the thickness of slab in inches; and b, 0.5791 inch, 
see pages 52 and 53. 



WOODEN BEAMS 



The accompanying tables are calculated on a 
basis of a maximum fibre strain of 1,000 pounds per 
square inch of material, or one-tenth of the break- 
ing strain of spruce. 

Floor loads as indicated. 

The length or span of rafter or floor beam in these 
tables is the clear length between supports. In 
rafters with collar beams, and without tie at the 
plate level, the length is to be taken as the length 
from the ridge to the point of connection of the 
collar beam, unless this is more than five times 
the distance on the rafter from the collar beam to 
the plate, in which case five times the latter distance 
is to be taken as the span in determining the dis- 
tance between centers. 



Distance between centres foi other thicknesses of beams or other unit 
strains will vary in direct proportion to the increase or decrease. Thus, 
3"x4"rafters at 1000 lbs. per square inch of material and 30 lbs. per square 
foot of roof may be put at 11" centres for 14 feet span, or 1 % of 7.3" as 
given for 2"x4"in the table. For 1500 lbs. instead of 1000 lbs. per 
square inch of material, a further increase of 50 per cent, may be made, 
and the centre distance becomes 16.5 inches. 



Span, feet, € 




8 


10 


12 


14 


16 


18 


20 


22 


24 


26 


28 


Size of 
Beam. 


Distance between centres of beams in inches. 


2x 4 
2x 5 

2x 6 
2x 7 
2x 8 
2x 9 


39 


5 


22.2 
34.7 


14 2 
22.2 
32.0 


9.9 
15.4 

22.2 
30.2 


7-3 
11.3 

16.3 
22.2 
29 


5-6 

8 7 

12-5 
17.0 
22 2 
28.1 
34.7 


6-9 

9.9 
13.4 

17.5 
22-2 
27.4 
39 5 


8.0 
10.9 
14.2 
18.0 

22.2 
32 


9.0 
11 7 
15.0 
18.4 

!6. 


9 9 
12.5 
15-4 

22.2 


10 6 
13 1 

19 




2x10 
2x12 














11 3 

16.4 



Where plastering is to be used, beams sho 
of as in table; and at 15" centres, depth i 
length in feet when beams are 2" thick. 





2x 4 


29.fi 


16.7 10.7, 7 4 
















~ 




,2x 5 




26.0 16.7 11.5 


8.5 














*o " 


CD 


2x 6 






24.0 16.7 


12.3 


9.4 












o § 


2x 7 








22.7 


16.7 


12.8 


10.1 












© 


2x 8 










21.8 


16.7 


13.2 


10 6 








«# 


2x 9 












21.1 


16.7 


13.5 


11.1 






~1 


e3 


2 x 10 














20.6 


16 7 


13.8 


11 6 




2x12 
















24.0 


19.8 


16 7 


14-2 





2x14 


















27.0 


22.7 


19.3 


16 7 


For plastering, when distance between centres equals 20", deptji of beam in. 




inches shoutd equal six-tenths of span in feet. 




2x 4 


23.7 


13.3 


8.5 


















51 -^ 




2x 5 




20.8 


13.3 


9.3 


















W3 


2x 6 






19.2 


13.3 


9.8 














js a 


2x 7 






26.1 


18.1 


13.3 


10 2 












■M£ 


© 


2x8 








23 7 


17 4 


13.3 


10 5 










U5 


2x 9 










22.0 


16.9 


13.3 


10-8 








9*5 


1 


2x10 












20.8 


16.5 


13.3 


11.1 


9.3 




2x12 














23.7 


19 2 


15 8 


13.3 


11 3 





2x14 
















26.1 


21.6 


18-2 


15.4 


13. 3 




For plastering, When distance between centres equals 16", depth of team 




in inches should equal six-tenths of span in feet. 



8 10 12 14 16 18 20 22 24 J86 28 



50 





Span, feet, 6 8 JO 12 14 16 18 20 22 


■S 


Sire of 


Distance between centres of beams in inches. 


2x 4 


11-9 




















c 
o 


2x 5 


186 


10.4 
















^ 


»j 


2x 6 


26.7 


15.0 


.9.6 














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o 


^ 


2x 7 




20.4 


13.1 


9.1 












fl 


<n 


1 

o 


2x 8 




26.7 


17.6 


11.8 


8.7 










% 


2* 9 






21.1 


15 


11.0 










» e 


s 


h) 


2^10 






26.7 


IS. 6 


13.7 


10.5 








^> «• 


J3 

H 




2x]2 








26.7 


19.6 


15.0 


11 8 


9.6 




<3 




2x14 










26.7 


20 5 


16.2 


13.1 


10 8 






.' 2 * 


12.7 


6-9 




















■fi 


2x 6 


17.7 


10.0 


6.4 














£ 


"5 


■* 


2x 7 


24.0 


13.6 


8.7 


6 












.e <» 


U) 


»© 


2x 8 




17 7 


11.3 


7 9 


5.8 










1»5 

i! 


O 




2x 9 




22.4 


14.3 


10.0 


7.3 


_5 6 








,C 


1 


2x10 




27.7 


17.7 


11 8 


9.3 


6.9 


5.5 






rt 


2x12 






26.5 


17.7 


13.0 


100 


7.9 


6 4 




|l B. 


£ 





2x14 








24 


17.7 


13 5 


10.7 


8.7 


7.2 


&3 




2*16 










23 1 17-7 14.0 11.3 


9.4 






. 


3x 5 


13.8 






















1 


3x 6 


20.0 


11.3 
















5 


*-j 


3x7 


27.2 


15 3 


9.8 














•X .• 


w 


o 
o 


3x 8 


35.6 


20-0 


12.8 


8.9 












If 
1 » 


o 


« 


3x 9 




25.2 


16.2 


11.3 


8.3 










XI 

g 


■0 

s 


3x10 




31 2 


20.0 


13. S 


10 2 


7.8 








3x12 






2S.8 


20.0 


14.7 


11.3 


8 9 






fe». 


£ 





3x14 








27.2 


20 15.3 12.6 


9.8 




S3 




3x16 










26.1120.0 15.8 


12 8 


10 5 






x 3x 5 


11.11 




, 














~ 


£> 


3x 6 


16. l! 9 
















£ 


j- .2 


■" 


3x 7 


21.912.3 


7-8 














CO 




© 


3x 8 


28.5 16 1 


10.2 


7.1 












fl 


2 ? 


e* 


3x 9 


36.0,20.13 


13.1 


9.0 


6 6 










*g to 


•§ 


3x10 


.. 25,1 


16.1 


11.1 


8.1 


6.3 








" ■§ 


J $ 


3x12 




36 


23.1 


16.1 


11 9 


9.0 


7-2 






~ a. 







3x14 






31 5 


21 8 


16.1 


12 3 


9-8 


7.8 




£ 


3x16 








28.5 


20.8 


16.1 


12 8 


10 2 


8.6 




a 


3x 6 


13-41 


















X! 


3x 7 


18.210-2 














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*J 


o 


3x 8 


23.7J13-4 


8.6 












3 


Q 

CO 


3x 9 


30.0 17 


10.8 


7-5 










O 


3x10 


37.1 


20.0 


13.4 


9.3 


6.8 








*■ « 


u 


o 


3x12 




30 


19 2 


13.4 


9.8 


7.5 






is 


i 


3x14 






2G 1 


18.2 13.4 


10.2 


8.0 




wj 


3x16 








23.7117.4 


13.4 


10 5 


8.6 







6 8 10 12 14 16 18 20 



CONCRETE MEMBERS 

How to Calculate Them When Subjected to 
Various Exterior Forces 



It would be a waste of time trying to learn to cal- 
culate concrete when in a mass state, that is, with- 
out reenforcement. When a member is reenforced, 
whatever kind of reenforcement may be used, the 
important point is to get this member to sustain 
safely the loads required with the least amount of 
concrete and steel reenforcement, thereby reducing 
the dead weight of the construction. For this pur- 
pose good judgment and a careful analysis of the 
member under consideration is necessary. 

Taking, for instance, a slab resting upon two par- 
allel beams or supports, imagine a rectangular slice 
cut from this slab at right angles to the beams and 
that this slice is then cut crosswise in the center or 
at half span. We then have to consider the cross- 
section at the weakest point of the concrete slab and 
have to find the amount of steel required over said 
span, within the considered slice, to carry the neces- 
sary load. This is done easily by the formulae on 
the accompanying plates. Naturally this slab is to 
be uniformly reenforced the same as at this con- 
sidered section. 

The calculations can then be readily computed, 
and if one were to use any kind of steel reenforce- 
ment other than the du Mazuel reenforcement the 
(see page 56) 



a £ s 



"gES 1 o o H «; , c a T3 

till * i ! i!i 3 £ 






■§S|| g. % 2 . ||i |° 



i**J 



3 s " -^3 3 « ° c c 



CO 



P § « 1"Hl8|sy SgSg ! 55?=g||Sg 

_ o P 

N u 

** w - Jo Ji-cj •*= c 'O 7J e "a 

2 -5 C | ! | S 3 I ~ 8 g | g - 5 V 

3 ■- » ,7 2 73 " ^ S u 0-2 «g H 5 u 3* 



™ 3 S3 E ° G g 5 

2«=5i ii » i « ii i « ii s 

W o g -" «■>&/)_}-■>£ E -a N ~ " 

< « ^ 



"I It" 3 . § a »=> §-E 8 .|g | s 

S, -"jg.o.sg S-S-'gS'l.Sc Sell 



Com c £ - o ! 
■3-g .2 g x) o ° ^ 

2Po 3 



U 
H 
O 




Considered Section, 




W =Uniformly distributed load per square foot. 



VS + * 
nS je 3 2 , ,,.) 



3WL 2 _ 3WL 2 

2 ' Z ~~W~ 



isnWf g 



54 




FORMULAE FOR BEAMS UNDER ORDINARY LOADINGS 




Method of Loading 


Maximum 
Bending 
Moment 


Maximum 
Load 

w 


Deflection 
d. 


Section 

modulus 

Z 


Length in feet 


Load in pounds 


Ft. Lb. 

M 


In. Lb. 
m 


Lb. 


In. 







L 


1 


WL 
8 


3 WL 
2 


2 2S 

3L 


5 Wl» 
384 EI 


3 WL 
2S 




w 






1 


I 1 




L 




WL 
6 


2 WL 


ZS 


Wl? 

60EI 


2WL 
S 






r^^Wi 


i 


c^ 


. *L , 




1 L 




WL 

2 


6WL 


ZS 

6L 


Wl J 
*E1 


6WL 
S 




: jr lllw i 




L 


>W 


WL 


12WL 


ZS 
12L 


Wl" 

3EI 


12WL 
S 




I 








L 


1 


WL 
4 


3 WL 


ZS 

3L 


Wl» 

48EI 


3WL 
S 


r*-jrH 




L 


1 


WA 

2 


6WA 


ZS 
6A 


Wa 
48EI* 

OlMa') 


6WA 




.* -A. 


J 


1 1 1 1 




L 




WA 

2 


6WA 


ZS 
6A 


Wa_ 
16EI* 

(l-2a)« 
Between 
Supports 


6WA 
S 


i 


. A A , 

4 r^ 


hw 


[J * i 




L 


WL 

3 


WL 


ZS 
•4L 


Wl 3 

UeT 


4WL 
S 




^^fetr^^ 




■» 




L 


i 


WL 
12 


WL 


ZS 
L 


3Wl s 
320 EI 


WL 
S 




^^iW i^-—" 


i 


' CTj 



L=span in feet; l=span in inches; W=*total load in pounds; E=modulus ol elasticity, 
I=moment of inertia; Z=section modulus ; S-safe 6tress on the extreme fibres of the beam 
section. In figuring deflections, all lengths must be expressed in inches. Small letters 
are used to represent inches, and capitals to represent feet. 



Delta ( A ) would represent the area of steel of the 
considered section while the distances c, b and f 
would refer to the ordinary theoretical center of 
gravity of the reenforcing member's cross section ; 
in the case of a round bar being used, the center of 
said bar being the center of gravity of its cross 
section. 

All members or parts of a concrete structure can 
be similarly analyzed. 

In the case of a column, the height being ex- 
pressed in feet and the radius of gyration in inches 
and used respectively as the numerator and denomi- 
nator of a fraction, such a radius of gyration should 
be chosen as will make the fraction equal one as 
nearly as possible. 

Definitions, together with the formulae, are given 
in the following pages for finding the sections re- 
quired under various conditions. Tables of proper- 
ties of Carnegie standard and special sections are 
also furnished, by the aid of which beams may be 
quickly and accurately proportioned. 

The use of these tables is fully explained by an 
example showing clearly how to determine beam 
sections required for both uniform and non-uniform 
loads. 



DEFLECTION 

Deflection is the tendency of a beam or slab to 
bend under loads. 

In some instances deflection, rather than absolute 
strength, may be the governing consideration in 



56 



determining the size of a beam to be used. For 
beams carrying plastered ceilings, for example, it 
has been found by practical tests that, if the deflec- 
tion exceeds 1/360 of the distance between the sup- 
ports, there is danger of the ceiling cracking. 

For deflections due to different systems of load- 
ing see plate of formulae for beams under ordinary 
loading's. 



REACTIONS 

Reaction is the force that supports the beam at a 
point. The sum of the reactions is always equal to 
the sum of the loads on the beam plus the weight 
of the beam itself. 

The accompanying figure represents a beam sup- 
ported on reaction 1 and reaction 2, and loaded 
in a uniform or non-uniform manner ; the loads 
being represented by W^ W 2 , W 3 , at distances A, 
B and C, from reaction t. 



Silk 



As the reaction at either support is the sum of 
the loads multiplied by their respective distances 



to the opposite reaction, and divided by the distance 
between the reactions, we have the following for- 
mulae : 

_ (Wj X A) + (W 2 X B) + (W 3 X C) 
R 2 — — — 

I, 

Rj = ( W t + W 2 + W 3 ) — R 2 



SHEAR 

Shear is that tendency to divide or cut a member 
or beam across its fibres when same is subjected to 
a force. At any section of the beam the shear equals 
either reaction minus the sum of the loads between 
that reaction and the section considered. The max- 
imum shear, which is always the one taken into con- 
sideration, is equal to the larger reaction. 



BENDING MOMENTS 

Bending moment is the force which, when 
exerted on a beam, will force the beam to bend or 
deflect. It is therefore necessary to find the maxi- 
mum bending moment to which a beam can be 
subjected. To find the maximum bending mo- 
ment, find the reactions and calculate the moments 
around each load, the greater moment being the 
maximum bending moment. 

Referring to the figure just now considered, we 
therefore have: 



Bending moment at W t 

equals Rj X A; 
Bending moment at W 2 

equals Rj X B — Wj X (B — A); 
Bending moment at W 3 

equals R r XC-W 2 X (C— B)— W 1 <(C— A); 
or, equals R 2 X (L — C); 
and so on. 



SECTION MODULUS 

The section modulus may be used in determin- 
ing the section required. The section modulus is 
equal to the maximum bending moment expressed 
in inch pounds "m", divided by the safe stress of the 
extreme fibres of the beam considered "S" ; in other 
words, divided by a safe working factor of the 
material used. 

Section Modulus = Z =— 



SAFE WORKING FACTORS 

A safe working factor is that fraction of ultimate 
load which is allowed per square inch of material. 

The working factors commonly used for steel are 
as follows : 

Private residences ; 20,000 pounds. 

Office and apartment buildings ; 16,000 pounds. 

Factory and heavy construction; 14,000 pounds. 

Rolling loads, as in bridges, etc. ; 12,000 pounds. 

Mills subjected to heavy shocks; 8,000 pounds. 



59 

EXPLANATION OF TABLES OF PROPERTIES OF 

CARNEGIE STANDARD AND SPECIAL 

I-BEAMS AND CHANNELS 

The tables on I-beams and channels are calcu- 
lated for all weights to which each pattern is rolled. 

Columns 8 and 9 give coefficients by means of 
which the safe, uniformly distributed load may be 
readily and quickly determined. To do this it is 
only necessary to divide the coefficient given by the 
span or distance in feet between supports. There- 
fore if, as will usually be the case, a section or beam 
is to be selected with a safe working factor of 12,500 
or 16,000 pounds, for carrying a certain load for a 
length of span already determined upon, it will only 
be necessary to ascertain the coefficient which this 
load and span will require and refer to the table, 
columns 8 and 9, for a section having a coefficient 
of this value. The coefficient is obtained by multi- 
plying the uniformly distributed load in pounds by 
the span length in feet. In case the load is con- 
centrated at the middle of the span, multiply the 
load by 2 and then consider it as uniformly dis- 
tributed. 

However, if other safe working factors are se- 
lected, and for other cases of loading, one has to ob- 
tain the maximum bending moment in inch pounds 
and divide it by the selected safe working factor. 
This will give the section modulus "Z", when refer- 
ence should be made to column 7 in table of prop- 
erties for the required beam. 

These tables have all been prepared with great 
care. No approximation has entered into any of the 
calculations, therefore the figures given may be 
relied upon as accurate.. 



PROPERTIES OF 


I-BEAMS 




i 


2 


3 


-4- 


S 


s 


7 


e 


& 


L 


1 
II 


11 




1. 

P 

1? 


Mom. of Inertia 

Neutral Axis 

m Perpendicular 

to Web at 

Center 


». Section Mod-. 

I^ulus Neutral 
Axis Perpendic- 
ular to Web at 
Center 


Coefficient of 
Strength for 
^Fiber Stress of 
"16,000 lbs. per 
sq. in. Used 
for Buildings 


Coefficient of 
Strength for 
rtKber Stress of 
Vl2,500 lbs. per 
so. in. Used 
lor Bridges 


24 


100.00 
95.00 
90.00 
85.00 
80.00 


29.41 
27.94 
26.47 
25.00 
23.32 


0.754 

o!631 

0.570 

0.500 


7.254 
7.192 
7.131 
7.070 
7.000 


2380.3 
2309.6 
2239.1 
2168.6 
2087.9 


198.4 
192.5 
186.6 
180.7 
174.0 


2115800 
2052900 
1990300 
1927600 
1855900 


1G53000 
1608900 
1554900 
1505900 
1449900 


20 


100.00 
95.00 
90.00 
85.00 
80.00 


29.41 
27.94 
26.47 
25.00 
23.73 


0.884 
0.810 
0.737 
0.663 
0.600 


7.284 
7.210 
7.137 
7.063 
7.000 


1655.8 
1606.8 
1557.8 
1508.7 
1466.5 


165.6 
160.7 
155.8 
150.9 
146.7 


1766100 
1713900 
1661600 
1609300 
1564300 


- 1379800 

1339000 

1298100 

1257200 

1222100 


20 


75.00 

70.00 

35.00 


22.06 

20.59 

19.08 


0.649 

0.575 

0.500 


6.399 

6.325 

6.250 


1268.9 

1219.9 

1169.6 


126.9 

122.0 

117.0 


1853500 

1301200 

124760O 


1057400 

1016600 

974700 


18 


70.00 

65.00 

60.00 

55.00 


20.59 

19.12 

17.65 

15.93 


0.719 

0.637 

0.555 

0.460 


6.259 

6.177 

6.095 

6.000 


921.3 

881.5 

841.8 

795.6 


102.4 
97.9 
93.5 

88.4 


1091900 

1044800 

997700 

943000 


853000 

816200 

779500 

736700 


15 


100.00 
95.00 
90.00 
85.00 

80.00 


29.41 
27.94 
26.47 
25.00 
23.81 


1.184 

1.085 

0.987 

0.889 

0.810 


6.774 
6.675 
6.677 
6.479 
6.400 


900 5 
872.9 
845.4 
817 8 
795.5 


120.1 
116.4 
112.7 
109.0 
106.1 


1280700 
1241500 
1202300 
1163000 
1131300 


1000600 
969900 
939300 
908600 
883900 


15 


75.00 

70.00 

65.00 

60.00 


22.06 

20.59 

19.12 

17.67 


0.882 

0.784 

0.686 

0.590 


6J94 

6.096 

6.000 


691.2 

663.6 

636.0 

609.0 


92.2 

88.5 

84.8 

81.2 


988000 

943800 

904600 

866 10O 


768000 

737400 

706700 

676600 


15 


55.00 

50.00 

45.00 

42.00 


16.18 

14.71 

13.24 

12.48 


0.656 

0.558 

0^60 

0.410 


5.746 

5.648 

5.550 

5.500 


511.0 

483.4 

455.8 

441.7 


68.1 

64.5 

60.8 

58.9 


726800 

687500 

648200 

628300 


567800 

537100 

506400 

490800 


12 


55.00 

50.00 

45.00 

40.00 


16.18 

14.71 

13.24 

11.84 


0^699 

0.576 

0.460 


5.612 

5.489 

5.366 

5.250 


321.0 

303.3 

285.7 

268.9 


53.5 

50.6 

47.6 

44.8 


570600 

539200 

507900 

478100 


445800 

421300 

396800 

373500 


12 


35.00 
31.50 


10.29 
9.26 


0.436 
0.350 


5.086 
5.000 


228.3 
215.8 


88.0 
36.0 


405800 
38370O 


317000 
299700 


10 


40.00 

35.00 

30.00 

25.00 


11.76 
10.29 
8.82 
7.37 


0.749 

0.602 

0.455 

0.310 


5.099 

4.952 

4.805 

4.660 


158.7 

146.4 

134.2 

122.1 


81.7 

29.3 

26.8 

24.4 


888500 

812400 

286300 

260500 


264500 

244100 

223600 

203500 





35.00 

30.00 

25.00 

21.00 


10.29 

7i35 
6.31 


0.732 

0.569 

0.406 

0.290 


4.772 

4.609 

4.446 

4.330 


111.8 

101.9 

91.9 

84.9 


24.8 
22.6 
20.4 
18.8 


265000 

241500 

217900 

2Q130O 


207000 

188700 

170800 

157300 


8 


25.50 

23.00 

20-50 

18.00 


7.50 

6.76 

6-03 

5.33 


0.541 

0.44S 

0.357 

0.270 


4-271 

4.179 

4.087 

4.000 


68.4 

64.5 

60.6 

56.9 


17.1 

16.1 

15-1 

14.2 


182500 

172000 

161600 

161700 


142600! 

134400 1 

126200 

118500 


7 


20.00 

17.50 

15.00 


5-88 

5.15 

4.42 


0.458, 

0.353 

0.250 


3^763 
3.660 


42.2 

39.2 

36.2 


12.1 

11.2 

lo:«4 


128600 

119400 

110400 


100400 

93300 

86300 


6 


17-25 

14.75 

12.25 


5.07 

4.34 

3.61 


0.475 

0.352 

0.230 


3.575 

3.452 

3.330 


26.2 

24.0 

21.8 


8-7 
8.0 
7.3 


93100 

85300 

77500 


' 72800 

66600 

60500 


5 


14.75 
12.25 
975 


4.34 

3.60 

2.87 


0.504 

0.357 

0.210 


3.294 

3-147 

3.000 


15.2' 

13-6 

12.1 


6.1 
5-4 
4.8 


64600 

58100 

51600 


50500 

45400 

40300 


4 


10.50 

9.50 

'8.50 

7.50 


3.09 

2.79 

2.50 

2.21 


0.410 

0.337 

0.263 

0.190 


2-880 

2.807 

2-733 

2.660 


7.1 
6.7 
6.4 
6.0 


3.6 
3.4 
3-2 
3.0 


38100 

36000 

33900 

31800 


29800 

28100 

26500 

24900 


3 


7.50 2.21 

6.50 1.91 

5.50 1.63 


0.361 

0.263 

0.170 


2.521 

2-423 

2.330 


2.9 
2-7 
2.5 


1.9 
1.8 
1.7 


20700 

19100 

17600 


16200 

15000 

13800 



w= 



CorC 



M: 



C or C 
8 



C or C ' =WL=8M 



8SZ 
= 12 







PROPERTIES OF 


CHANNELS 




1 


2 


3 


-f- 


3 


6 


7 


a 


9 


II 

IT 


|1 
•f 2 


a on 
11 

Id 


1^1 


I, 


Mom. of Inertia 

Neutral Axis 

Perpendicular 

to Web at 

Center 


Sectiun Mod- 
ulus Neutral 
AxisPerpendic- 
, ular to Web 
at Center 


Coefficient of 
Strength for 
Fiber Stress of 
16,000 lbs. per 
sq. in. Used 
for Buildings 


Coefficient of 
Strength for 
Fiber Stress of 
12,500 lbs. per 
sq. in. Used 
for Bridges 


e-> 


^ 




> 


I 


Z 


c 


c 




55.00 


16.18 


0.818 


3.818 


430.2 


57.4 


611900 


478000 




50.00 


14.71 


0.720 


3.720 


402.7 


53.7 


572700 


447400 


15 


45.00 


13.24 


0.622 


3.622 


375.1 


50.0 


533500 


416800 


40.00 


11.76 


0.524 


3.524 


347.5 


46.3 


494200 


386100 




35.00 


10.29 


0.426 


3.426 


320.0 


42.7 


455000 


355500 




33.00 


9.90 


0.400 


3.400 


3.12.6 


41.7 


444500 


347300 




40.00 


• 11.76 


0.758 


3.418 


197.0 


32.8 


350200 


273600 




35.00 


10.29 


0.636 


3.296 


179.3 


29.9 


318800 


249100 


12 


30.00 


8.82 


0.513 


3.173 


161.7 


26.9 


287400 


224500 




25.00 


7.35 


0.390 


3.050 


144.0 


24.0 


256100 


200000 




20.50 


6.03 


0.280 


2.940 


128.1 


21.4 


227800 


178000 




35.00 


10.29 


0.823 


3.183 


115.5 


23.1 


246400 


192500 




30.00 


8.82 


0.676 


3.036 


103.2 


20.6 


220300 


172100 


10 


25.00 


7.35 


0.529 


2.889 


91.0 


18.2 


194100 


151700 




20.00 


5.88 


0.382 


2.742 


78.7 


15.7 


168000 


131200 




15.00 


4.46 


0.240 


2.600 


66.9 


13.4 


142700 


111500 




25.00 


7.35 


0.615 


2.815 


70.7 


15.7 


167600 


130900 


9 


20.00 


5.88 


0.452 


2.652 


60.8 


13.5 


144100 


112600 


15.00 


4.41 


0.288 


2.488 


50.9 


11.3 


120500 


94200 




13.25 


3.89 


0.230 


2.430 


47.3 


10.5 


112200 


87600 




' 21.25 


6.25 


0.582 


2.622 


47.8 


ll.fr 


127400 


99500 




18.75 


5.51 


0.490 


2.530 


43.8 


11.0 


•116900 


91300 


8 


16.25 


4.78 


0.399 


2.439 


39.9 


10.0 


106400 


83200 




13.75 


4.01 


0.307 


2.347 


36.0 


9.0 


96000 


75000 




11.25 


3.35 


0.220 


2.260 


32.3 


8.1 


86100 


67300 




19-75 

17?25 


5.81 


0.633 


2.513 


33.2 


9.5 


101100 


79000 




5.07 


0.528 


2.408 


30.2 


8.6 


92000 


-71800 


7 


14.75 


4.31 


0.423 


2.303 


27.2 


7.8 


82800 


64700 




12.25 


3.60 


0.318 


2.198 


24.2 


6.9 


73700 


57500 




9.75 


2.85 


0.210 


2.090 


21.1 


6.0 


66800 


52200 




15.50 


4.56 


0.563 


2.283 


19.5 


6.5 


69500 


54300 


6 


13.00 


3.82 


0.440 


2.160 


17.3 


5.8 


61600 


48100 


10.50 


3.09 


0.318 


2.038 


15.1 


5 


53800 


42000 




8.00 


2.38 


0.200 


1.920 


13.0 


4.3 


46200 


36100 




11.50 


3.38 


0.477 


2.037 


10.4 


4.2 


44400 


34700 


6 


9.00 


2.65 


0.330 


1.890 


8.9 


3.5 


37900 


29600 




6.50 


1.95 


0.190 


1.750 


7.4 


3.0 


31600 


24700 




7.25 


2.13 


0.325 


1.725 


4.6 


2.3 


24400 


19000 


4 


6.25 


1.84 


0.252 


1.652 


4.2 


2.1 


22300 


17400 




5.25 


1.55 


0.180 


1.580 


3.8 


1.9 


20200 


15800 




6.00 


1.76 


0.362 


1.602 


2.1 


1.4 


14700 


11500 


o 


5.00 


1.47 


0.264 


1.504 


1.8 


1.2 


13100 


10300 




4.00 


1.19 


0.170 


1.410 


1.6 


1.1 


11600 


910O 



C or C 



M: 



CorC 
8 



CorC = WL=8M= 



8SZ 
12 



W = Safe load in pounds uniformly distributed; L = 5pan in feet ; M = Bending 
moment in foot pounds; S= Safe working stress; C and C'= Coefficients; Weights 
in heavy print are standard; others are special. 

NOTE.— The two tables give the properties of Carnegie Standards. 



TYPICAL EXAMPLE 



When the floor beams have been laid out to suit 
the requirements or location of the ground, the 
floor design may then be quickly and accurately 
arrived at by following the various steps as outlined 
below. 



SLABS 

The spans and loads being given, we will refer 
to the diagrams or curves of safe loads. 

Three panels are to carry 250 pounds per square 
foot, uniformly distributed on ten and eleven foot 
spans : 

On diagram of curves for slabs reenforced with 
Standard No. 22 du Mazuel reenforcement, we fol- 
low the safe live load vertical line marked 250 until 
we reach the clear span horizontal line marked II, 
and find that a 2-inch slab is too small while a 2.y 2 - 
inch slab weighing 34.05 pounds per square foot 
would answer the purpose and is therefore chosen. 

Two panels are to carry 400 pounds per square 
foot, uniformly distributed on 6-foot spans : 

On diagram of curves for slabs reenforced with 
Standard No. 26 du Mazuel reenforcement, we find 
that a 2-inch slab would fully answer the purpose. 
However, as the top surface of a factory floor usu- 
ally should be level, we will choose a similar slab 
of 2.y 2 inches, weighing 32.51 pounds per square foot. 



MAIN BEAMS 

Main beams are those beams that do not directly 
carry the floor panels, but which support the secondary 
beams carrying these panels. 



SECONDARY BEAMS 

The loading on the secondary beams is uniform, 
and each beam carries half of its adjacent panel's 
live and dead loads. 



RULE FOR SELECTING MAIN OR SECONDARY 
BEAMS 

When selecting a member, whether for a main or a 
secondary beam, the first consideration should be 
standardization and lightness of weight. In other 
words it is preferable to choose a standard size and 
of the lightest weight: — a 24" I @ 80 pounds per 
lineal foot of beam is preferable to a 20" I @ 80 pounds 
where the greater depth is not objectionable; a 20" I 
@ 65 pounds to a 1 5" I @ 80 pounds, and so on. 



BEAM "BD" 

This beam, therefore, carries a load of 250 pounds 
plus 34.05 pounds by 18 feet by 5 feet. 

W = 284.05 X 18 X 5 = 25,565 pounds; 
and the section modulus, from the table of general 

formulae. 

(see page 66) 



PLAN OF A FACTORY FLOOR 

Shaded part to be loaded to 400 pounds per square 
foot, unshaded part to 250 pounds per square foot. 

Columns or piers are shown by circles, main girders 
or beams by heavy lines, and secondary girders or 
beams by ordinary solid lines. 

The safe working factor to be 14,000 pounds per 
square inch of steel. 




Q<> 



will give 

3X25,565 X 18 



^ 49.30 
2 X 14,000 
Reference to column y of table of properties of 
I beams, shows that a 12" I @ 50 pounds would be 
required. It will be found best, however, to choose 
a standard and therefore a 15" I @ 42 pounds is 
selected . 

Ri and R 2 for this beam will be equal to half the 
load W plus half of the weight of the selected beam 
itself. 

Rj = R 2 = 12,782.5 + 378 = 13,160.5 pounds. 



BEAMS "DF" AND "CE" 

These two beams are alike and carry the same 
loads, viz : 250 pounds plus 34.05 pounds by 18 feet 
by 5.5 feet. 

W = 284.05 X 18 X 5.5 = 28,121 pounds 
3X28,121X18 



=54.23 



2X14,000 
calling for 15" I @ 42 pounds each. 

Rj = R 2 = 14,060.5 + 378 = 14,438.5 pounds. 



BEAM "IJ" 

For this beam the loading is 250 pounds plus 34.05 
pounds by 18 feet by 11 feet. 

W== 284.05X18X11 = 56,242 pounds. 
3X56,242X18 



2X14,000 



-=108.47 



calling for a 20" I @ 65 pounds. 

Rj = R 2 =28,121 +585 = 28,706 pounds. 



BEAM "GH" 

This beam supports two half panels of unequal 
loading. One is 250 pounds plus 34.05 pounds by 
18 feet by 5 feet, the other is 400 pounds plus 32.51 
pounds by 18 feet by 3 feet. 

284.05X18X5 = 25,565 
432.51 X 18X3 = 23,356 
W = 48,921 pounds. 

3X48,921X18 

Z = = 94.348 

2X14,000 

calls for a 15" I @ 80 pounds, but it is better to 

choose a 20" I @ 65 pounds. 

Rj = R 2 = 25,045.5 pounds. 



BEAM "KL" 

The load here is 400 pounds plus 32.51 pounds by 

18 feet by 6 feet. 

W = 432. 51X18X6 = 46, 711 pounds. 

3X46,711X18 

Z = = 90.086 

2X14,000 

calling for a 15" I @ 80 pounds, but it is better to 

choose a 20" I @ 65 pounds. 

Rj = R 2 = 23,940.5 pounds. 



BEAM "AC" 

The load here is 400 pounds plus 32.51 pounds by 
18 feet by 3 feet. 

W == 432.51 X 18 X 3 = 23,356 pounds. 

3X23,356X18 

Z = = 45.04 

2X14,000 
calling for a 15" I @ 42 pounds. 

R 1 = R 2 = 12,056 pounds. 



BEAM "AB" 

This beam carries two reactions : R x of beam "K 
L", and R 3 of beam "GH" at points "K" and "G", 
6 and 12 feet away from "A". 

(23,940.5X6) + (25,045.5X12) 

R _= =20,191 pounds, 

22 F 

Rj =28,795 pounds. 

Maximum bending moment will be found to be : 

m = (20,191X10X12) inch pounds. 

20,191X10X12 

Z= = 170.06 

14,000 

calling for a 24" I @ 80 pounds. 

And as all reactions include the beams : 

R t =29,675 pounds, 

R 2 = 21,071 pounds. 



BEAM "CD" 

This beam carries three reactions : R 2 of beam "K 
L", R 3 of beam "IJ", and R 2 of beam "GH", at points 



69 



"I/", "I", and "H", 6, n and 12 feet from "C". 

„ (23,940.5 X 6)+(28,706 X 11 )+(25.045.5 X 12) 

-K 2 = =34,544 lbs. 

22 

m=(34,544Xll — 25,045.5X1) X 12 = 354,938.5X12 

354,938.5X12 

Z = = 304.23 

14,000 

This section modulus is larger than that given by 
the biggest standard stock beam rolled, therefore it 
will have to be divided by two and the two beams 
'corresponding to said arrived section moduli will be 
the beams chosen for "CD", one of these carrying as 
much as "AB", and the other as much as "EF", there- 
fore two 24" I beams @ 80 pounds will be selected. 

This "CD" would then give as reactions the sum 
of similar reactions of "AB" and."EF", and includ- 
ing the weight of the beams as total reactions at 
"C" and "D". 

Rj =44,908 pounds. 

R 2 = 36,304 pounds. 



BEAM "EF" 

This beam carries only one reaction at center; 

and from the table of general formulae is had: 

3X28,706X22 

Z= =135.33 

14,000 

calling for a 20" I @ 80 pounds, but we will make it 
a 24" I @ 80 pounds. 

Total reactions will include the beams : 



Rj = R 2 = 15,233 pounds. 



70 



COLUMNS OR PIERS 



Pier "A", as all others, has to be so designed as 
to safely support the reactions at that point, viz. : 
41,731 pounds; pier "B" will therefore have to sup- 
port safely 34,231.5 pounds; pier "C", 71,402.5 
pounds; pier "D", 63,903.0 pounds; pier "E", 29,- 
671.5 pounds; and pier "F", 29,671.5 pounds. 



FOUNDATIONS 

Undoubtedly sound rock is the ideal foundation 
for structures. However, comparatively seldom 
does one find such a foundation in the locality where 
a building is to be erected, and one has to be satisfied 
by whatever ground there is at the spot, going deep 
enough to safeguard against frost, — that is one 
foot or more below the frost line. 

As all earths do not sustain loads in the same 
proportion, the foundation has to be brought to a 
tapering or spreading base, as required, to cover 
enough ground and thereby allow it to sustain the 
loads to be piled above. 

Below is a table of the sustaining powers of 
different earths. 



BEARING POWER OF SOILS 




Kind of material 


Bearing power in tons 
per square foot 




Min. 


Max. 


Rock — the hardest — in thick layers, in native bed 


200 
25 
15 
5 

4 

2 

8 
4 
2 
0.5 


30 


Rock equal to best brick masonry 


20 
10 




6 




4 




2 




10 




6 


Sand, clean, dry 

Quicksand, alluvial soils, etc 


4 

1 



71 



Hereunder are given examples of various pressures 
on different kinds of foundations for important work 
bv eminent engineers. 



Pressure on Foundations.. 

structure. soil. tons per- sq. foot. 

Cleveland Viaduct .. .. Blue clay .. .. 1*0 to 1 -7 

Busigny Bridge .. .. U nstable sand 1-8 

„ Arched Bridge . . Yellowsandy clay.. ., .. 2'lto'2 8 

■Chimney, Newcastle .. Compact clay 1-5 

Kurtenberg Bridge . . . Compact sand .. 2*3 to 2*9 

Chimpey, New York ... Wet sand* ... ... .. .. 4*0 

Viaduct, Pont de Jour . Coarse gravel . . „ 44 

Brooklyn Bridge .. .Sand.. ..._ 4-0 

,, ,, .. Compact st oney clay .. .. 5-5 

Nantes Bridge Sand* 6-78 

Bordeaux Bridge .. .. Compact sand and gravel .. 7*36 to 8 '17 

Washington Memorial .. Clay and sand* 3 to 9* 

* Settlement took place. 



ARCHES 



Many ways, graphical and theoretical, have been 
employed hitherto to design arches, but few of these 
ways, however, seem to be adaptable to all cases, 
and especially to concrete and reenforced concrete. 

To cover reenforced concrete arches, therefore, it 
will be found advisable to combine both the graphi- 
cal and theoretical. 

As an arch may yield under the pressure to which 
it is subjected, either by the slipping of certain of 
its parts of contact upon one another, or by their 
turning over upon the theoretical edges of one 
another; and as these two conditions involve the 
whole question of its stability, the important point 
is to keep all of the forces within certain limits or 
bounds of said arch so as to prevent such slipping 
or overturning. These forces, commonly regarded 
as the line of pressure of an arch, should always be 
within the middle third of said arch ring so as to 
do away with all or most of the tension forces, and 
the arch ring from the crown or keystone to abut- 
ments should first of all be approximately estab- 
lished. This ring or vousoir would correspond to 
the thickness of a du Mazuel reenforced slab greatly 
increased. 

All loads, live and dead, should be reduced to the 
same standard as that of the material of the arch 
itself, masonry or concrete, to one hundred and 
forty (140) pounds per cubic foot; and so as to 
simplify computations the arch should be considered 



to be a slice one foot in thickness, all other widths 
being a reproduction of the slice, thus all superficial 
measurements in the computations will represent 
cubic contents. 

To avoid a long explanation of the subject an 
example is given and worked out, and if one follows 
it step by step, any arch, however complicated, can 
be readily and accurately computed. 



EXAMPLE OF A du MAZUEL ARCH 

The accompanying' plate gives the design of an 
arch made of concrete reenforced by the du Mazuel 
system of reenforcement. This arch, which has to 
span clear one hundred and fifty (150) feet, is to be 
used as a country highway capable of sustaining a 
live and uniform load of six hundred (600) pounds 
per lineal foot and must be so designed as to resist 
the waters at their highest- flood level. 

In choosing a location for the bridge, reference 
must be had to the ground upon which it is to stand, 
both as to the abutments on the banks of the river, 
and as to the piers which have to stand in the 
waterway. Piers should be avoided if possible, as 
it has been found dangerous to encroach upon the 
original width of rivers. Sound beds of primi- 
tive or of horizontally stratified rock, and well 
compact chalk, are unexceptional as substrata 
for bridge construction. Sound, hard, clayey gravel 
through which water will not rise if tried with a 
head equal to that which it may be subjected 
to in floods, forms an excellent base, and so do hard 
(see page 75) 



Sf </) cu 

£ * u 

2 s * 





n ^ / 




1 


IF/ / ^ 


1 M 










/w K 


i-i'i- \ 




P' : ¥\ 3 


$ 




fefg 3 






f f 2 2 






r ; I 1 ■» 


s 




M o "2 


j 




1- 1 
1 m 

1 \ 
\ 


{ 

1 

■ : 1 














/ 






y Jk 


1 « 




1 r#' 




<h 


l# 1 





c §Ml 




and not easily soluble or stiff clays, when in beds 
of considerable depth or thickness : but soft, soluble 
clays ; loose, sandy gravel, and sand in any uncom- 
bined state, and subject to the action of the water, 
are not to be trusted. It must, too, be always borne 
in mind that the contraction of the waterway, which 
a bridge is almost certain to effect in a greater or 
less degree, tends to induce a more severe action 
upon the bed of the river than its substance can have 
been previously subjected to. 

Of course, facility of approach to a bridge must 
not be overlooked in selecting a site upon which to 
build it, and it is essential to the desirability of a 
site that the approaches can be made direct and 
easy. The economical consideration must determine 
whether it is better to take up a more difficult 
position for the bridge, to obtain less expensive 
approaches, or to take up the best site for the 
bridge, and bestow whatever labor may be requisite 
to make suitable approaches for it. 

Generally, however, in towns and cities, and for 
the most part throughout old settled countries and 
in commercial districts, bridges must be built where 
bridges are wanted, and a bridge must be adapted 
to its site. 

The situation of the bridge being determined, a 
careful survey should be made of it and the pro- 
posed line or lines of approach, and of the river 
over which the bridge is to be built, for the whole 
length, at least, of the reach upon which it is to be 
placed. The results of the survey should be laid down 
as a map or plan presenting an ichnographic outline 
or representation upon a horizontal section of lines 



raised vertically from every point of the surface of 
the ground ; for it must be obvious upon an inspec- 
tion of the diagram, that a plan of the site from A 
to B, or from one side of the river to the other laid 
down from measurements made along the bending 
line of the surface, would give a much greater length 
than a plan upon a horizontal section of lines raised 
vertically from the surface of the ground to the 
straight and horizontal line AB. The plan required 
is such as this latter description indicates and as 
we have shown it. The irregularities of the ground 
must also be represented in diagrams such as the 
above, so that in arranging, designing, and estimat- 
ing the intended work, the engineer may have the 
means before him of ascertaining the exact dimen- 
sions yielded by the site in every direction. Such 
a diagram as the above is termed a section and it 
should show, in addition to the irregularities of the 
ground, the various substances of which the ground 
is composed, and the thickness vertically of each 
substance as far as the object in view renders it 
necessary to ascertain — that is, until proper sub- 
stance upon which to work has presented itself. 
This may be obtained by boring with augers made 
for the purpose. 

The abutments and abutment footings or piers 
are then carried to sound strata. The arch itself 
or intrados is laid out so as to give ample headway 
without raising the general highway level, and the 
springs of the arch are started above the highest 
flood level. The abutment footings, being most ex- 
posed to the currents of the water, are laid out of 
du Mazuel oleaginous concrete with a face smooth- 
trowel finished. 

The abutments have been erected to the lines 
of theoretical haunches and thoroughly covered with 
heavy coats of asphaltum so as to allow for expansion, 
and the work is now readv for the arch erection 



proper. Before going further into the work, however, 
consider the general detail design of this arch. 

The bridge considered without its spandrel fill- 
ing, roadbed, railings, moldings, etc., etc., — that is, 
the arch proper, — the depth of which at crown has 
been defined by the following empirical formula: 

l/Radius + half span . .. 

-r 40^ 

4 

and the depth of skewbacks having been determined 

as double the depth of the crown, a most satisfactory 

rule for concrete arches, — is laid out to scale, and is 




[ 1 i I I 1 1 I I II I 1 I I i 

» 9 160 300 30o 400 S60 600 700 



c 
o 


G' 


A' 


A'xG" 


(A'xG')s 


H 


M' 

or 

A'xG'xH 


M' s 


M's 


(A'xG')s 


1 


2.5 


9.03 


22.58 


22.58 


1.25 


28.23 


28.23 


1.25 


2 


5 


9.03 


45.15 


67.73 


5. 


225.75 


253.98 


3.75 


3 


5 


9.5 


47.5 


115.23 


10. 


475. 


728.78 


6.33 


4 


5 


9.5 


47.5 


162.73 


15. 


712.5 


1441.28 


8.86 


5 


5 


to. 


50. 


212.73 


20. 


1000. 


2441.28 


11.48 


6 


5 


11. 


55. 


267.73 


25. 


1375. 


3816.28 


14.26 


7 


5 


10.5 


52.5 


320.23 


30. 


1575. 


5391.28 


16.84 


8 


5 


12. 


60. 


380.23 


35. 


2100. 


7491.28 


19.7 


9 


5 


13.5 


67.5 


447.73 


40. 


2700. 


10191.28 


22.76 


10 


5 


11.5 


57.5 


505.23 


45. 


2587.5 


12778.78 


25.29 


11 


5 


14. 


70. 


575.23 


50. 


3500. 


16278.78 


28.3 


12 


5 


16.5 


82.5 


657.73 


55. 


4537.5 


20816.28 


31.65 


13 


5 


14. 


70. 


727.73 


60. 


4200. 


25016.28 


34.38 


14 


5 


17.5 


87.5 


815.23 


65. 


5687.5 


30703.78 


37.66 


15 


5 


21.5 


107.5 


922.73 


70. 


7525. 


38228.78 


41.43 


16 


2.5 


24.5 


61.25 


98 3.98 


73.75 


4517.19 


42745.97 


43.44 


17 


6.54 


23. 


150.42 


1134.4 


78.27 


11773.37 


54519.34 


48.06 



divided into any number of stations, the more the 
better, seventeen (17) being considered sufficient in 
this instance to demonstrate the theory. Over this 
arch are laid out imaginary strips of masonry equal 
to the six hundred (600) pounds per lineal foot, and 
in this case, equal to 4.29 at their smallest height. 

Having prepared a table as shown here we then 
proceed to fill it in for each and every station. 

Equating the moment around the theoretical span 
point, that is around the third of the skewback in 
conjunction to center of gravity of the considered 
part of the arch, we have for the present case 77.18 
minus 48.06 from last column of above table, or a 
distance equal to 29.12 feet toward center of span, and 



get for the uniform horizontal pressure or thrust for 
each section, expressed in cubic feet of concrete 

Y X ( A 1 X G 1 ) n X b 



Theoretical rise 
29 12X1,134.40X1.16 



a, 183.8 



32.37 

and expressed in pounds, 1,183.8 X 140 = 165730 
pounds, b being the distance in inches between the 
plane of apexes of upward forces in du Mazuel re- 
enforcement to lower surface of arch or soffit — b 
should never exceed 1.1666 inches, or approximately 
mean depth of du Mazuel reenf or cement, but should 
be chosen proportionately to the cross area of said 
reenforcement. 

The line of pressure can now be readily plotted 
as follows : 

Along a horizontal line passing through the theo- 
retical rise point or through the center of the arch 
proper at the crown, and from the center on the 
line mark off the distances of the last column in table, 
1.25, 3.75, 6.33, etc., etc., this being to same scale as 
arch. From ends of these distances draw perpen- 
dicular lines on which mark off to some convenient 
scale distances equal to amounts in the fourth 
column, 22.58, 67.73, 115.23, etc., etc., and, from 
these points draw the uniform or common horizontal 
thrust expressed in cubic feet to same convenient 
scale. From these last points are drawn the ex- 
tended hypothenuses of the several triangles form- 
ing the pressure curve as shown. This curve should 
always be within the middle third of the arch proper. 

The next steps are the laying of steel angle 
trusses, laying of reenforcement, laying of con- 
crete for arch proper, embedding of the steel angle 
trusses in concrete, filling in of earth to proper 
grades, and then finishing the bridge to architectural 
lines. 



SHEET PILING 



In many cases the du Mazuel reenforcement made 
out of plain sheets may be used satisfactorily as a 
light, efficient sheet pile. Of course, in cases where 
boulders are present trouble would be experienced 
with the sheets crumpling, but in sandy, marshy soils, 
or in soft, wet ground, like mud, silt, swamp, quick- 
sand, loam, or soft clay, where great difficulty is 
encountered in excavating, in keeping out water, in 
excluding the soft material from the excavation and 
in preventing the undermining of adjacent struc- 
tures, the Standard No. 22 du Mazuel reenforce- 
ment will offer the only perfect interlocking, light, 
stiff, easily driven, waterproof economical sheet pile. 




81 



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04 







HEATING SYSTEMS 



Relative to heating a building, the engineer, archi- 
tect or builder should invariably avoid the use of 
the hot air furnace. Indirect hot water is the 
ideal system. The cost of installation and mainte- 
nance, however, makes it generally prohibitory. 
Direct hot water is the practical, commercial way 
of warming the smaller buildings. The steam heat- 
ing system commonly used for large buildings is 
preferable only in the case of indirect radiation. 

The following rule for determining the size of 
radiation needed for a given room will be found to 
be very satisfactory: 

Add the area of the glass surface in the room to 
one quarter of the exposed wall surface, and to this 
add from 1/55 to 3/55 of the cubical contents (1/55 
for rooms on upper floors, 2/55 for rooms on lower 
floors and 3/55 for large halls) ; then for steam mul- 
tiply by 0.35, and for hot water by 0.50. 



83 



EXAMPLE 

An upper story room 20X12X10 feet with glass 
exposure of 48 feet, % of wall exposure (two sides 
exposed) 320 feet = 80, 1/55 of 2,400 = 44. 
48 + 80 + 44=172 X 0.35 = 60.20 

60.20 feet is the radiation required for that room. 



84 



FIRE AND LOAD TESTS 

of 

DU MAZUEL REENFORCEMENTS 

made at 

TESTING STATION 

OF THE CITY OF NEW YORK 



[^ «HT 


r 








j 


If*" 


■ 


i 



IN compliance with the Building Code of New York, a fire, water 
and load test was made of the du Mazuel reenforcement. 
Thedu Mazuel reenforcement sheets were laid on top cords from 
girder to girder ; the concrete being dumped and spread over them. 



85 




The under side was then plastered, forming a ceiling of three 
clear spans of 6 feet, 3 l /z feet and 9 l 4 feet respectively, the 6 
foot slab being 3)4 inches in thickness and the considered section. 



86 




The entire building was constructed of concrete and provided 
with six chimneys and eight draft openings at the bottom, with 
two openings closed by doors for putting in the fuel. 




A uniform load of 150 to 160 pounds per square foot was laid on 
top of the du Mazuel construction, and steel level rods 
were placed thereon for determining deflections. 




An oak fire was started and kept up for four hours, so as to give 
a uniform heat throughout that time of from 1,750 to 1,860 
degrees Fahrenheit. 




At the end of the four hours, the New York City Fire Department 
came with hose and fire engine, and directed a 1 X ' ncn stream 
of water, having the tremendous force of between 60 and 80 
pounds per square inch against the top and bottom of the du 
Mazuel construction, which had reached a white heat. 




The next day, when the whole building had cooled, it was noticed 
that although the ceiling plastering under the du Mazuel construc- 
tion had been injured 



91 




the beams were in a worse condition, the concrete around them 
having cracked, broken and fallen in many places. These beams 
were of the standard kind as used in most first-class, fireproof 
buildings. 




Additional piles of pig iron were then heaped upon the du Mazuel 
construction, subjecting it to a stress, as required by the Building 
Code, of 600 to 650 pounds per square foot, uniformly distributed. 
Although a great part of the plastering had been burned out, or 
washed out by the high pressure stream of water, the deflection 
was then only a trifle over 2% inches, and when this great load 
was removed the permanent deflection was only 2-& inches, 
showing that the loads had scarcely any effect whatever. 



93 



PERMITS 

As a result of the fire, water and load tests shown 
and explained on preceding pages, the Bureaus of 
Buildings of the various Boroughs of the City of New 
York, granted general permits or approval of the du 
Mazuel reenforcements, for general use in fireproof 
or other buildings within the City of New York. 



TABLES, ETC. 



WEIGHT OF A CUBIC FOOT OF SUBSTANCES 



Aluminum 162 

Anthracite, solid, of Pennsylvania 03 

" broken, loose 54 

" " moderately shaken 58 

" heaped bushel, loose (80) 

Ash, American, white, dry 38 

Asphaltum 87 

Brass (Copper and Zinc), cast 604 

" rolled 524 

Brick, best pressed 160 

" common, hard 125 

" soft, inferior 100 

Brickwork, pressed brick 140 

" ordinary 112 

Cement, hydraulic, ground, loose, American Rosendale . 56 

" " " Louisville . 50 

" "" " " English, Portland ... 90 

Cherry, dry 42 

Chestnut, dry 41 

Clay, potters', dry 119 

" in lump, loose 63 

Coal, bituminous, solid ■ 84 

" broken, loose 49 

heaped bushel, loose (74) 

Coke, loose, of good coal 26.3 

" heaped bushel (40) 

Copper, cast 542 

rolled 548 

Earth, common loam, dry, loose 76 

" moderately rammed 95 

as a soft flowing mud . 108 

Ebony, dry >jq 

Elm, dry .... ok 

Flint .162 



97 



WEIGHT OF A CUBIC FOOT OF SUBSTANCES 
Continued 



Glass, common window •..,.. 157 

Gneiss, common 168 

Gold, cast, pure, or 24 carat 1204 

" pure, hammered * 1217 

Grain, at 60 lbs. per bushel 48 

Granite 170 

Gravel, about the same as sand, which see. 

Gypsum (plaster of paris) 142 

Hemlock, dry 26 

Hickory, dry 63 

Hornblende, black 203 

Ice 68.7 

Iron, cast 460 

" wrought, purest 485 

" " average 480 

Ivory . . . . 114 

Lead 711 

Lignum Vitse, dry 83 

Lime, quick, ground, loose, or in small lumps .... 63 

" - " " " thoroughly shaken .... 76 

" " " u per struck bushel .... (QQ) 

Limestones and marbles 168 

41 " loose, in irregular fragments . » 06 

Magnesium 10® 

Mahogany, Spanish, dry ...» 63 

" r Honduras, dry 36 

Maple, dry 49 

Marbles, see Limestones. 

Masonry, of granite or limestone, well dressed .... 166 

" ■" mortar rubble ...*....- 164 

" " dry " (well scabbled) 138 

" " sandstone, well dressed 144 

Mercury, at 32° Fahrenheit 649 

Mica 183 

Mortar, hardened 103 

Mud, dry, close 80 to HO 

Mud, wet, fluid, maximum 120 

Oak, live, dry. 60 



98 



WEICHT OF A CUBIC FOOT OF SUBSTANCES 
Continued 



Oak, white, dry 60 

" other kinds 32 to 45 

Petroleum 55 

Pine, white, dry 25 

" yellow, Northern 34 

" " Southern 45 

Platinum 1342 

Quartz, common, pure 165 

Rosin 69 

Salt, coarse, Syracuse, N. Y 45 

" Liverpool, fine, for table use 49 

Sand, of pure quartz, dry, loose 90 to 106 

" well shaken . . . 99 to 117 

perfectly wejt 120 to 140 

Sandstones, fit for building 151 

Shales, red or black 162 

Silver 655 

Slate 175 

Snow, freshly fallen 5 to 12 

" moistened and compacted by rain 15 to 50 

Spruce, dry 25 

Steel , 490 

Sulphur .... 125 

Sycamore, dry 37 

Tar 62 

Tin, cast . 459 

Turf or Peat, dry, unpressed 20 to 30 

Walnut, black, dry 38 

Water, pure rain or distilled, at 6O0 Fahrenheit .... 32^ 

"sea 64 

Wax, bees 60.6 

Zinc or Spelter 437.6 

Green timbers usually weigh from one-fifth to one-half more than dry. 
All loads are expressed in pounds. 



UNITED STATES STANDARD GAUGE FOR SHEET 
AND PLATE IRON AND STEEL 

Adopted as Standard by American Railway Master Mechanics Asso- 
ciation and Association of American Steel Manufacturers 



feSo 

is 

S5 


Approximate 

Thickness in 

Fractions 

of an Inch 


f ' s i-s 


III 
III 


Weight per 
Square Foot in 
Pounds Avoir- 
dupois, Iron 


Weight per 
Square Foot in 
Pounds Avoir- 
dupois, Steel 


Weight per 
Square Meter 

in Kilo- 
grammes Steel 


If 


0000000 


1—2 


.5 


12.70 


20. 


20.4 


99.601 


0000000 


000000 


15—32 


.46875 


11.91 


18.75 


19.125 


93.376 


000000 


00000 


7—16 


.4375 


11.11 


17.50 


17.85 


87.151 


00000 


0000 


13—32 


.40625 


10.32 


16.25 


16.575 


80.926 


0000 


000 


3-8 


.375 


9.53 


15. 


15.3 


74.701 


000 


00 


11—32 


.34375 


8.73 


13.75 


14.025 


68.476 


00 





5—16 


.3125 


7.94 


12.50 


12.75 


62.251 





1 


9—32 


.28125 


7.14 


11.25 


11.475 


56.026 


1 


2 


17-64 


.265625 


6.75 


10.625 


10.8375 


52.913 


2 


3 


1—4 


.25 


6.35 


10. 


10.2 


49.800 


8 


4 


15-64 


.234375 


5.95 


9.375 


9.5625 


46.688 


4 


5 


7—32 


.21875 


5.56 


8.75 


8.925 


43.575 


5 


6 


13—64 


.203125 


.5.16 


8.125 


8.2875 


40.463 


6 


7 


3—16 


.1875 


4.76 


7.5 


7.65 


37.850 


7 


8' 


11-64 


.171875 


4.37 


6.875 


7.0125 


34.238 


8 


9 


5-32 


.15625 


3.97 


6.25 


6.375 


31.125 


9 


10 


9—64 


.140625 


3.57 


5.625 


5.7375 


28.013 


10 


11 


1-8 ] 


.125 


3.18 


5. 


5.1 


24.900 


11 


12 


7—64 


.109375 


2.78 


4.375 


4.4625 


21.788 


12 


13 


3—32 


.09375 


2.38 


3.75 


3.825 


18.675 


13 


14 


5-64 


.078125 


1.98 


3.125 


3.1875 


15.563 


14 


15 


9-12S 


.0703125 


1.79 


2.8125 


2.86875 


14.006 


15 


16 


1—16 


.0625 


1.59 


2.5 


2.55 


12.450 
14.205 


16 


17 


9-160 


.05625 


1.43 


2.25 


2.295 


17 


18 


1-20 


.05 


1.27 


2. 


2.04 


9.960 


18 


19 


7—160 


.04375 


1.11 


1.75 


1.785 


8.715 


19 


20 


3-80 


.0375 


0.953 


1.50 


1.53 


7.470 


20 


21 


11-320 


.034375 


0.873 


1.375 


1.4025 


6.848 


21 


22 


1—32 


.03125 


0.794 


1.25 


1.275 


6.225 


22 


23 


9—320 


.028125 


0.714 


1.125 


1.1475 


5.603 


23 


24 


1—40 


.025 


0.635 


1. 


1.02 


4.980 


24 


25 


7—320 


.021875 


0.556 


.875 


.8925 


4.358 


25 


26 


3—160 


.01875 


0.476 


.75 


.765 


3.735 


26 


27 


11-640 


.0171875 


0.437 


.6875 


.70125 


3.424 


27 


28 


1—64 


.015625 


0.397 


.625 


.6375 


3.113 


28 


29 


9—640 


.0140625 


0.857 


.5625 


.57375 


2.801 


29 


30 


1—80 


.0125 


0.818 


.5 


.51 


2.490 


30 


31 


7—640 


.0109375 


0.278 


.4375 


.44625 


2.179 


81 


32 


13—1280 


.01015625 


0.258 


.40625 


.414875 


2.023 


82 


33 


3—320 


.009375 


0.238 


.375 


.8825 


1.868 


83 


3-1 


11—1280 


.00859375 


0.218 


.34875 


.350625 


1.712 


84 


35 


5—640 


.0078125 


0.198 


.3125 


.31875 


" 1.556 


85 


30 


9—1280 


.00703125 


0.179 


.28125 


.286875 


1.401 


86 


87 


17—2560 


.006640625 


0.169 


.265625 


.2709875 


1.823 


87 


38 


1—160 


.00625 


0.159 


.25 


.255 


1.245 


88 



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102 ____________ — — — _. 

Dr du Mazuel's Logarithms of numbers to ioo, of primes under 

noo and of certain other numbers, to sixty-one places. 

For ordinary calculations take only first few places. 



0.00000,00000,00000,00000,00000,00000,0(3000,00000,00000,00000,00000,000000 

0.30102,99956,63981,195x1,37388,94724,49302,67681,89881,46210,85413,104275 

>.477i2,i2547,l9662,43729,5O279,O3255,ii530,920Oi,28864,i9o69,5864b,298t>50 
0.60205,999^3,27962,39042,74777,89448,98605,35363 ,79762,924»i,7 *26,ao8549 
0.6989 7,00043 , 3 6018,804 78,62611,05275,5 0697,32 318,10115,5^789,14586^93725 



o.778i5,i2503 ) 83643,63a 5 o',87667,97979.6o833,59 6 83,i874S,65a8o 5 44o6i^o293» 
O.84509,8o4oo ) i4256.830 7 i,22i6-2,58592,636i9,34835,72396 > 32396,54o65,03«'35 
0.90308,99869,91943,58564,12166,84173,47908,03045,69644,38632,56239,312824 
0.95424,25094,39324,87459,00558,06510,23061,84002,57728,38139,17296,597313 
T. 00000,00000,00000,00000,00000,00000,00000,00000,000 oo,ooooo,ooooo> 000000 



Logarithms 



.20411,99826,55924,78085,49555,78897,97210,70727 59$z5,84843,4i65z,4i7 9 8 
:.-3°44,892r 3^78273,92854,01698,9432-8,33703,00075,67378,42504,63973 803685 
.25527,25051,03306^06980,37947,01234,72364,5168447609,84350,02769,70x587 
•27875,36009,52828,96153,63334,75756.92931,79511,29337,39449,75989,068189 
:»3 0102,99956,6 398 1,19 52 1,37 388,94724,49302,6768 1,8988 1,462 10^8541 3,104275 



3222 1,92947,3 39 1 9.-26800,72441, 61847 ,75 1 50,26837,01260,5 i466,i27i3,'3350o6 

34242,26808,22206,23596,39388,65967, 5 i726,84748,9»07i,92856.i6359,©69665 

1. 36172,78360,17592,87886,77771,1225 1, 18954,9697 5,1 1034,33609,61 882^56055 

i.3'8o2 1,12417,1 1606,02293,62445,87428,59438,95046,98508, 57702,i4887«6ii48o 

39794,00086,72037,609 57,25221,105 51,01394,64636,20137,07578,29x73,79x451 



4H97,33479,7° 81 7,96442,02440,52666,82 145,75979,19069,84917,68131,116184 16 
i-43 l 3 6 ,3764i,5 8 987,3 1,8s >5°837,09765,34592,7 6 o°3,86592,572o8,75944,895969 Z7 
1.44715,80313,42219,22113,96940,48041,62224,70199,52159,24818,24891,144899 " 
1.46239,79978,98956,08733,28467,62969,25499,12542,94417,88715,38410,653969 

4 7712,12547,19662,43729,50279,0325 5,1 1 530,92001,28864,19069,1:8648.198656 



..041 39,2685 1, 58225,O4075,or999,7i243,o»424,i7o67,oai90^46645, 30945,965390 1 
1.07918, 12460,47624,82772,25056,92704, 10136,27 365,08627,1 i4^i,29474,5 7 zO ^ " 
.11394,33523,06836,76920,65051,57942,32843,08297,29188,38706,847*8,011910 
.14612,803 56.78238,02592,59551, 533i7,i29'22pi547,62277,78667,39478,i4o624 14 
.17609,12 590,55681,14208,12890,08 530,62228,243 19^8982. 72858,73235,19438a 15 

" 16 
»7 
18 



49136,10938,34272,67966,67041,00118,41572,13037,01558,30418,46559,383498 

1.50514,99783,19905,97606,86944,73622,46513,38405,49407,31054,27065,521373 

" i^j 1 , 39398,77887,47804,52278,74498,13955,0^68,31054,65714,89594,264047 

53H7 ,89170,42255.12375,39087,89052,83005,67757,57259,887x5^49386,90795.9 

54406,80443.50275,63549,84773,63868,14316,67153,82514.86185,68651,932075 



1.536,0,25007,67287,26501,75335,95959,11667,19366,37491,30560,88122,805862 
1.56820,17240,66994,99680,84506,89539,12944,79829,72690,16631,25466,176799 
1 57978.35966,16810,15675,00723,70481,4223447193,19218,85660,61402,172463 
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*-97358»962 34,27256,90834,2297 5,10551,79624,82320,81816,02752,59675,858750 
2.97634,99790,03273,41875,01137,75925,22039,01622,95145,98964,50857,664310 
2 >979°9>290o6,38326,4o8'53,29393. 47717,31227,47302,58220,10598,20494,3657 10 
2.98542,64740,83001,67359,77060,21186,62711,98227,26427,50112,13308,635787 



9/1 2. 9872i,92299,68oo4i8628o,3 1 389,065 36,25 140,4053 1, 99480, 84889, 06195,03 1 8 34 
977 2.98989,45637,18773,07091,48028,11052,34926,25914,08310,84838,41813,133125 
9832.99255,35178,32135,62274,96349,24741,43755,19748,99290,01915,16629,651606 
991 2.99607,36544,85275,32236,44343,78815,42086,41325,12663,22812,08187,848418 
_997 2.99869,51583,11655,71988,13717,02813,27230,27091,29009,56252,34578,237114 
1009 3.00389,11662,36910,52171,52813,16509,5^886,55201,95652,55260,09846,3^2385 
1013 3'0°56o,94453,6o28o,42845,oi6i7,2oo70,22i65,o8630,76662.o6266,67962,258954 
1019 3-00817,41840,06426,39490,49899,22311,83296,76922,24936,36781,15542,425256 
1021 3.00902,57420,86910,24724,81480,36966,37851,03031,35315,99655^5437,518936 
1031 3.01325,86652,83516,54690,96644,09013,44583,24998,28006,59445,12546,30173c 



1033 

*°39 
1049 
105 j 

106: 



3.01410,03215,19620,57904,40100,62744,77060,74356,51400,55338,40683,272162 
3.0.1661,55475,57177,41240,21010,01361,62758,71828,97066,20300,27455,551333 
3.02077,54881,93557,85990,72007,63899,91741,19141,56191,40400,29271,212173 
3.02160,27160,28242,22008,37688,89097,91 687,94575i 69660,00863, 2 3290,07 150c 
571,53839,^340,66612,28844,7399078^53,18778 , 5 6 167,59 546, 12209 ,8 374 6: 



3.02653,32645,23296,75697,14741,94622,85093,72551, 33664,50701,42150,259662 
1069 3.02897,77052,o877,8,oi749,oi456,79857 > 36936 / 27594,4S925,ooS24,96999,t32959S 
1087 3-03 6 22,9544o,86294,53992,62573, 76344,441 1571246,06239,23536,422 16,49471c 
1091 3.03782,47505,88341,87761,10634,29318,59826,96526,11482,20421,01725,763338 
1093 3.03862,01619,49702,79226,92555,27640,43892,4947676830,67575,50087,010561 
1097 3.04020,66275,7471 i,i'322 1,54832,40551, 60744,802 36,80562,48547,7753 1,00941s 



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1 


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33 


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i 


2 


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17 


34 


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3 


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35 


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18 


36 


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5 


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37 


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3 


6 


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19 


38 


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7 


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39 


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4 


8 


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1-8 


20 


40 


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5-8 




9 


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41 


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5 


10 
11 


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21 


42 
43 


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6 


12 


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3-16 


22 


44 


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11-16 




13 


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45 


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7 


14 


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23 


46 


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.15 


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47 


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8 


16 


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1-4 


24 


48 


•75 


3-4 




17 


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49 


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9 


18 


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25 


50 


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19 


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51 


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IO 


20 


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5-16 


26 


52 


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13-16 




21 


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53 


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ii 


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27 


54 


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23 


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55 


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12 


24 


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3-8 


28 


56 


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7-8 




2-5 


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57 


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13 


26 


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29 


58 


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27 


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59 


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14 


28 


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7-16 


30 


60 


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29 


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61 


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15 


30 


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3* 


62 


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63 


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32 


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CO ^J-O t^ 
vo vo vo vo 

q q q q 


VO On N vo 

O0 On - N 
vo voO O 

q q q q 


u 
2 


«N<***ieH' a J<o 




ttset 


^'-»«wt-'f 



o 

I 

W 
o 

B 



S 



■5 

M 
o 

P3 

s 

EH 

O 

o 



o 

H 



^ 


U">00 « 3£ 

e>5 oo oo oo 
o\ q» q* on 


r«»0 W© 
u->t>»00 ON 

00 00 00 00 
ON ON ON ON 


ON ON On ON 

ON ON ON ON 


vo t^oo o 

On On On O 

ON ON ON O 


V 

o 


t^OO ON i 
On ON ON O 

00 00 00 ON 


COO On 0» 
CM co -3-0 
O O O O 

ON ON ON ON 


O On CM vo 
t>.00 O M 

O O 5 m 

ON ON ON ON 


CM ^t vo 

On On On 


0> 


00 m -3- t^ 
co voo t>» 

00 00 00 00 


O COO ON 
ON O -i M 
ii CM M N 

00 00 00 00 


<M VO00 i-i 

■<*■ mo oo 

NNNfl 
00 00 00 00 


3;no 

On O cm 

N CO CO 

00 00 00 


00 


VO00 i-i rf 

CO ro co co 
f«» r>» r^. r>. 


t^ O coO 
vo r^oo on 
coco co co 
t-» r^ r^ t>» 


On CM voOO 
O CM CO -* 


O t^OO 

-<* ^ ^tf- 




£- 


t^OO ON n 
rj" rj- ^(- vo 
O O O O 


COO On M 
N CO "vt-O 
m m m vo 

o o o o 


o on « m 

t^OO O M 

m mo o 

o o o o 


00 M Tj" 

N Tj- vo 

o o o 
o o o 


co 


00 -*"«*■ t^ 
co mo t--. 

O O O O 


O COO ON 
ONO « N 
o t>» *>. l>. 
lomiom 


n moo n 
•<$• mo oo 
r-» r-* r>. r>» 
tovo vo vo 


■* t^ o 

ONO M 
t^»00 00 
VO vo vo 




£ 


moo w -* 
O i-i co <*■ 

00 00 00 00 

"t "t "*" T 


r~- O COO 

VOl>»00 ON 

oo oo oo oo 
*t T 't "* 


On CM voOO 
O M CO ■<*■ 
ON On O On 

t *t "t t 


O t^OO , 

ON On On\ 

1 1 "t - 


^ 


1 rj" t^ O 

r^oo on i-i 

ON ON ON O 

CO to CO "tf- 


COO On CM 
C4 CO "3-o 


O On CM vo 
I>-00 O i 

o o « •-• 

<tf- <tf- -^- <tf- 


00 -i '4- 
CS "tf- vo 

't 4" 4" 


CO 


00 H^N 
CO mO t>. 

CO CO CO CO 


O coO On 

i CM CM CM 
CO CO CO CO 


cm moo ** 
•*t vno oo 

CM N CM CM 

CO CO CO CO 


ON O M 
CM CO CO 
CO CO CO 


CN 


1/100 11 *fr 

O i co rf 
CO CO co co 

N N N N 


!>. O COO 

mt^oo on 

co co co CO 

NN N « 


On CM voOO 
O CM CO <* 
<^ "* "^ '* 
CM CM CM CM 


O t^CO 
N M N 




~ 


Jt^OO ON « 


COO ON CM 
CM CO "tf-O 


O ON CM VO 

t^OO O i-i 

vo mo o 


oo ii <d- 

CM rj- vo 

«o o o 


o 


00 n ■* t^ 
to mo r-» 

VO O O O 
°. °. °. °. 


coO On 

q q q q 


CM VOOO M 

<!t mo 00 
i>. t~-» r>. r>» 

q q q q 


ON O CM 

l>.00 00 

q q q 


X 
u 




S5SSS5- 


U5J-— ftf,«lff]Ori;i-l 


*»- 



3 s. 



EIGHT TRIANGLES 




TO FIND A. 



GIVEN. 


FORMULA. 


GIVEN. 


FORMULA. 


b,c 
b, a 


b 
tan A - - 

. A b 

sin A = - 

a 


c,b 
c, a 


c 

COt .4 = r 



c 

cos .4 = - 

a 



b^c 


cot B = - 
c 


c, 6 


tan 5 = J 




6, a 


cos B = - 
a 


c, a 


sin 5 =- 
a 



A,b 
A,c 



b 

' sin A 




TO FIND 6. 



b 
" cos B 



A,c 
A, a 



b = c tan A 
b = a sin A 



B,a 

B,c 



b = a cos B 
b — c cot B 



TO FIND c. 



A, a 
A,b 



c = a cos A 
c — b cot .4 



£, a 
B,b 



c = a sin 5 
c — b tan 5 



OBLIQUE TRIANGLES 




TO FIND a, 6, c. 


TO FIND A, B, C. 


GIVEN. 


FORMULA. 


GIVEN. 


FORMULA. 


A, b, C 
A,B,c 
A,B,b 


_ 6 sin C 
*~ sin A 
c sin A 
sinB 
__ bsinB 
— Sin A 


a, b, C 
A,b,c 
A, a, b 


. . b sin C 
sin A = 

. „ C sin 4 
sin B = r 



. _ a sin .4 
sin C = j- — 





TO FIND A, B t C. 

s = i(o + 6 + c). 



GIVEN. 


FORMUK2E. 






a, c, s 
a, 6, s 


. . > /(s — a) (s — c) 
■»^ = |/ ac 

sm^ = |/ a& 




sm jc - |/ &c 



114 



s 

1 


SINE. 




O' 


10' 


20' 


30' 


40' 


60' 


60' 







0.00000 


0.00291 


0.00582 


0.00873 


0.01164 


0.01454 


0.01745 


89 


1 


0.01745 


0.02036 


0.02327 


0.02618 


0.02908 


0.03199 


0.03490 


88 


2 


0.03490 


0.03781 


0.04071 


0. 04362 


0.04653 


0.04943 


0.05234 


87 


3 


0.05234 


0.05524 


0.05814 


0.06105 


0.06395 


0.06685 


0.06976 


86 


4 


0.06976 


0.07266 


0.07556 


0.07846 


0.08136 


0.08426 


0.08716 


85 


5 


0.08716 


0.09005 


0.09295 


0.09585 


0.09874 


0.10164 


0.10453 


84 


6 


0.10453 


0.10742 


0.11031 


0.11320 


0.11609 


0.11898 


0.12187 


83 


7 


0.12187 


0.12476 


0.12764 


0.13053 


0.13341 


0.18629 


0.13917 


82 


8 


0.13917 


0.14205 


0.14493 


14781 


0.15069 


0.15356 


0.15643 


81 


9 


0.15643 


0.15931 


0.16218 


0.16505 


0.16792 


0.17078 


0.17365 


80 


10 


0.17365 


0.17651 


0.17937 


0.18224 


0.18500 


0.18795 


0.19081 


79 


11 


0.19081 


0.19366 


0.19052 


0.19937 


0.20222 


0.20507 


0.20791 


78 


12 


0.20791 


0.21076 


0.21360 


0.21644 


0.21928 


0.22212 


0.22495 


77 


13 


0.22495 


0.22778 


0.23062 


0.23345 


0.23627 


0.23910 


0.24192 


76 


14 


0.24192 


0.24474 


0.24756 


0.25038 


0.25320 


0.25601 


0.25882 


75 


15 


0.25882 


0.26163 


0.26443 


0.26724 


0.27004 


0.27284 


0.27564 


74 


16 


0.27564 


0.27843 


0.28123 


0.28402 


0.28680 


0.28959 


0.29237 


73 


17 


0.29237 


0.29515 


0.29793 


0.30071 


0.30348 


0.80625 


0.30902 


72 


18 


0.30902 


0.31178 


0.31454 


0.31730 


0.32006 


0.32282 


0.32557 




19 


0.32557 


0.32832 


0.33106 


0.33381 


0.33655 


0.33929 


0.34202 


70 


20 


0.34202 


0.34475 


0.34748 


0.35021 


0.35293 


0.35565 


0.35837 


69 


21 


0.35837 


0.36108 


0.36379 


0.36650 


0.36921 


0.37191 


0.37461 


68 


22 


0.37461 


0.87730 


0.37999 


0.38268 


0.38537 


0.38805 


0.39073 


67 


23 


0.39073 


0.39341 


0.39608 


0.39875 


0.40142 


0.40408 


0.40674 


66 


24 


0.40674 


0.40939 


0.41204 


0.41469 


0.41734 


0.41998 


0.42262 


65 


25 


0.42262 


0.42525 


0.42788 


0.43051 


0.43318 


0.43575 


0.43837 


64 


26 


0.43837 


0.44098 


0.44359 


0.44020 


0.44880 


0.45140 


0.45399 


63 


27 


0.45399 


0.45658 


0.45917 


0.46175 


0.46433 


0.46690 


0.46947 


62 


28 


0.46947 


0.47204 


0.474C0 


0.47716 


0.47971 


0.48226 


0.48481 


61 


29 


0.48481 


0.48735 


0.48989 


0.49242 


0.49495 


0.49748 


0.50000 


60 


30 


0.50000 


0.50252 


0.50503 


0.50754 


0.51004 


0.51254 


0.51504 


59 


31 


0.51504 


0.51753 


0.52002 


0.52250 


0.52498 


0.52745 


0.52992 


58 


32 


0.52992 


0.53238 


0.53484 


0.53730 


0.53975 


0.54220 


0.54464 


57 


33 


0.54464 


0.54708 


0.54951 


0.55194 


0.55436 


0.55678 


0.55919 


56 


34 


0.55919 


0.56160 


0.56401 


0.56641 


0.56880 


0.57119 


0.57358 


55 


35 


0.5^358 


0.57596 


0.57833 


0.58070 


0.58307 


0.58543 


0.58779 


54 
53 


36 


0.58779 


0.59014 


0.59248 


0.59482 


0.59716 


0.59949 


0.60182 


37 


0.60182 


0.60414 


0.60645 


0.60876 


0.61107 


0.61337 


0.61566 


52 

51 


38 


0.61566 


0.61795 


0.62024 


0.62251 


0.62479 


0.62706 


0.62932 


39 


0.62932 


0.63158 


0.63383 


0.63608 


0.63832 


0.64056 


0.64279 


50 


40 


0.64279 


0.64501 


0.64723 


0.64945 


0.65166 


0.65386 


0.65606 


49 


41 


0.65600 


0.65825 


0.66044 


0.66262 


0.66480 


0.66697 


0.66913 


48 
47 
46 
45 


42 


0.66913 


0.67129 


0.67344 


0.67559 


0.67773 


0.67987 


0.68200 


43 


0.68200 


0.68412 


0.68624 


0.68835 


69046 


0.69256 


0.69466 


44 


0.69466 


0.69675 


0.69883 


0.70091 


0.70298 


0.70505 


0.70711 




60' 


60' 


40' 


30' 


20' 


10' 


<y 


S 




COSINE. 


1 



115 



1 


COSINE. 




«* 


0' 


10' 


20' 


30' 


40' 


60' 


60' 







1.00000 


1.00000 


0.99998 


0.99996 


0.99993 


0.99989 


0.99985 


89 


1 


0.99985 


0.99979 


0.99973 


0.99966 


0.99958 


0.99949 


0.99939 


88 


2 


0.99939 


0.99929 


0.99917 


0.99905 


0.99892 


0.99878 


0.99863 


87 


3 


0.99863 
6.99756 


0.99847 


0.99831 


0.99813 


0.99795 


0.99776 


0.99756 


86 


4 


0.99736 


0.99714 


0.99692 


0.99668 


0.99644 


0.99619 


85 


5 


0.99619 


0.99594 


0.99567 


0.99540 


0.99511 


0.99482 


0.99452 


84 


6 


0.99452 


0.99421 


0.99390 


0.99357 


0.99324 


0.99290 


0.99255 


83 


7 


0.99255 


0.99219 


0.99182 


0.99144 


0.99106 


0.99067 


0.99027 


82 


8 


0.99027 


0.98986 


0.98944 


0.98902 


0.98858 


0.98814 


0.98769 


81 


9 


0.98769 


98723 


0.98676 


0.98629 


0.98580 


0.98531 


0.98481 


80 


10 


0.98481 


0.98430 


0.98378 


0.98325 


0.98272 


0.98218 


0.98163 


79 


11 


0.98163 


0.98107 


0.98050 


0.97992 


0.97934 


0.97875 


0.97815 


78 


12 


0.97815 


0.97754 


0.97692 


0.97630 


0.97566 


0.97502 


0.97437. 


77 


13 


0.97437 


0.97371 


0.97304 


0.97237 


0.97169 


0.97100 


0.97030 


76 


14 


0.97030 


0.96959 


0.96887 


0.96815 


0.96742 


0.96667 


0.96593 


75 


15 


0.96593 


0.96517 


0.96440. 


0.96363 


0.96285 


0.90206 


0.96126 


74 


16 


0.96126 


0.96046 


0.95964 


0.95882 


0.95799 


0.95715 


0.95630 


78 


17 


0.95630 


0.95545 


0.95459 


0.95372 


0.95284 


0.95195 


0.95106 


72 


18 


0.95106 


0.95015 


0.94924 


0.94832 


0.94740 


0.94646 


0.94552 


71 


19 


0.94552 


0.94457 


0.94361 


0.94264 


0.94167 


0.94068 


0.93969 


70 


20 


0.93969 


0.93869 


0.93769 


0.93667 


0.93565 


0.93462 


0.93358 


69 


21 


0.93358 


0.93253 


0.93148 


0.93042 


0.92935 


0.92827 


0.92718 


68 


22 


0.92718 


0.92609 


0.92499 


0.92388 


0.92276 


0.92164 


0.92050 


67 


23 


0.92050 


0.91936 


0.91822 


0.91706 


0.91590 


0.91472 


0.91355 


66 


24 


0.91355 


0.91236 


0.91116 


0.90996 


0.90875 


0.90753 


0.90631 


65 


25 


0.90631 


0.90507 


0.90383 


0.90259 


0.90133 


0.90007 


0.89879 


64 


'26 


0.89879 


0.89752 


0.89623 


0.89493 


0.89363 


0.89232 


0.89101 


63 


27 


0.89101 


0.88968 


0.88835 


0.88701 


0.88566 


0.88431 


0.88295 


62 


28 


0.88295 


88158 


0.88020 


0.87882 


0.87743 


0.87603 


0.87462 


61 


29 


0.87462 


0.87321 


0.87178 


0.87036 


0.86892 


0.86748 


0.86603 


60 


30 


0.86603 


0.86457 


0.86310 


0.86163 


0.86015 


0.85866 


0.85717 


59 


31 


0.85717 


0.85567 


0.85416 


0.85264 


0.85112 


0.84959 


0.84805 


58 


82 


0.84805 


0.84650 


0.84495 


0.84339 


0.84182 


0.84025 


0.83867 


57 


33 


0.83867 


0.83708 


0.83549 


0.83389 


0.83228 


0.83066 


0.82904 


56 


34 


0.82904 


0.82741 


0.82577 


0.82413 


0.82248 


0.82082 


0.81915 


55 


35 


0.81915 


0.81748 


0.8i580 


0.81412 


0.81242 


0.81072 


0.80902 


54 


36 


0.80902 


0.80730 


0.80558 


0.80386 


0.80212 


0.80038 


0.79864 


53 


37 


0.79864 


0.79688 


0.79512 


0.79335 


0.79158 


0.78980 


0.78801 


52 


38 


0.78801 


0.78622 


0.78442 


0.78261 


0.78079 


0.77897 


0.77715 


51 


39 


0.77715 


0.77531 


0.77347 


0.77162 


0.76977 


0.76791 


0.76604 


50 


40 


0.76604 


0.76417 


0.76229 


0.76041 


0.75851 


0.75661 


0.75471 


49 


41 


0.75471 


0.75280 


0.75088 


0.74896 


0.74703 


0.74509 


0.74314 


48 


42 


0.74314 


0.74120 


0.73924 


0.73728 


0.73531 


0.73333 


0.73135 


47 


43 


0.73135 


0.72937 


0.72737 


0.72537 


0.72337 


0.72186 


0.71934 


46 


44 


0.71934 


0.71732 


0.71529 


0.71325 


0.71121 


0.70916 


0.70711 


45 




60' 


50' 


40' 


30' 


20' 


10' 


0' 


T 










SINE. 









116 



i 


TANGENT 






0' 


10' 


20' 


30' 


40' 


60' 


60' 







0.00000 


0.00291 


0.00582 


0.00873 


0.01164 


0.01455 


0.01746 


89 


1 


0.01746 


0.02036 


0.02328 


0.02619 


0.02910 


0.03201 


0.03492 


88 


2 


0.03492 


0.03783 


0.04075 


0.04366 


0.04658 


0.04949 


0.05241 


87 


8 


0.05241 


0.05533 


0.05824 


0.06116 


0.06408 


0.06700 


0.06993 


86 


4 


0.06993 


0.07285 


0.07578 


0.07870 


0.08163 


0.08456 


0.08749 


85 


5 


0.08749 


0.09042 


0.09335 


0.09629 


0.09923 


0.10216 


0.10510 


84 


6 


0.10510 


0.10805 


0.11099 


0.11394 


0.11688 


0.11983 


0.12278 


83 


7 


0.12278 


0.12574 


0.12869 


0.13165 


0.13461 


0.13758 


0.14054 


82 


8 


0.14054 


0.14351 


0.14648 


0.14945 


0.15243 


0.15540 


0.15838 


81 


9 


0.15838 


0.16137 


0.16435 


0.16734 


0.17033 


0.17333 


0.17633 


80 


10 


■0.17633 


0.17933 


0.18233 


0.18534 


0.18835 


0.19136 


0.19438 


79 


11 


0.19438 


0.19740 


0.20042 


0.20345 


0.20648 


0.20952 


0.21256 


78 


12 


0.21256 


0.21560 


0.21864 


0.22169 


0.22475 


0.22781 


0.23087 


77 


13 


0.23087 


0.23393 


0.23700 


0.24(108 


0.24316 


0.24624 


0.24933 


70 


14 


0.24933 


0.25242 


0.25552 


0.25862 


0.26172 


0.26483 


0.26795 


75 


15 


0.26795 


0.27107 


0.27419 


0.27732 


0.28046 


0.28360 


0.28675 


74 


16 


0.28675 


0.28990 


0.29305 


0.29621 


0.29938 


0.30255 


0.30573 


73 


17 


0.30573 


0.30891 


0.31210 


0.31530 


0.31850 


0.32171 


0.32492 


72 


18 


0.32492 


0.32814 


0.33136 


0.33460 


0.83783 


0.34108 


0.84433 


71 


19 


0.34433 


0.34758 


0.35085 


0.35412 


0.35740 


0.36068 


0.36397 


70 


20 


0.36397 


0.36727 


0.37057 


0.37388 


0.37720 


0.38053 


0.88386 


69 


21 


0.38386 


0.38721 


0.39055 


0.39391 


0.39727 


0.40065 


0.40403 


68 


22 


0.40403 


0.40741 


0.41081 


0.41421 


0.41763 


0.42105 


0.42447 


67 


23 


0.42447 


0.42791 


0.43186 


0.43481 


0.43828 


0.44175 


0.44523 


66 


24 


0.44523 


0.44872 


0.45222 


0.45573 


0.45924 


0.46277 


0.46681 


65 


25 


0.46631 


0.46985 


0.47341 


0.47698 


0.48055 


0.48414 


0.48773 


64 


26 


0.48773 


0.49134 


0.49495 


0.49858 


0.50222 


0.50587 


0.50953 


63 


27 


0.50953 


0.51820 


0.51688 


0.52057 


0.52427 


0.52798 


0.53171 


62 


28 


0.53171 


0.53545 


0.53920 


0.54296 


0.54674 


0.55051 


0.55431 


61 


29 


0.55431 


0.55812 


0.56194 


0.56577 


0.56962 


0.57348 


0.57735 


60 


30 


0.57735 


0.58124 


0.58513 


0.58905 


0.59297 


0.59691 


0.60086 


59 


31 


0.60086 


0.60483 


0.60881 


0.61280 


0.61681 


0.62083 


0.62487 


58 


32 


0.62487 


0.62892 


0.63299 


0.63707 


0.64117 


0.64528 


0.64941 


57 


33 


0.64941 


0.65355 


0.65771 


0.66189 


0.66608 


0.67028 


0.67451 


56 


34 


0.67451 


0.67875 


0.68301 


0.68728 


0.69157 


0.69588 


0.70021 


55 


35 


0.70021 


0.70455 


0.70891 


0.71329 


0.71769 


0.72211 


0.72654 


54 


36 


0.72654 


0.73100 


0.73547 


0.73996 


0.74447 


0.74900 


0.75855 


53 


37 


0.75355 


0.75812 


0,76272 


0.76733 


0.77196 


0.77661 


0.78129 


52 


38 


0.78129 


0.78598 


0.79070 


0.79544 


0.80020 


0.80498 


0.80978 


51 


39 


0.80978 


0.81461 


0.81946 


0.82434 


0.82923 


0.83415 


0.83910 


50 


40 


0.83910 


0.84407 


0.84906 


0.85408 


0.85912 


0.86419 


0.86929 


49 


41 


0.86929 


0.87441 


0.87955 


0.88473 


0.88992 


0.89515 


0.90040 


48 


42 


0.90040 


0.90569 


0.91099 


0.91633 


0.92170 


0.92709 


0.93252 


47 


43 
44 


0.93252 


0.93797 


0.94345 


0.94896 


0.95451 


0.96008 


0.96569 


46 


0.96569 


0.97133 


0.97700 


0.98270 


0.98843 


0.99420 


1.00000 


45 




scy 


40' 


30 / 


20' 


10' 


0' 


T 

a 




COTANGENT 







117 



1 


COTANGENT 






0' 


10' 


20' 


30' 


40' 


60' | 60' 







00 


343.77371 


171.83540 


114.58865 


85.93979 


68.75009 57.28996 


89 


1 


57.28996 


49.10388 


42.96408 


38.18846 


34.36777 


81.24158 28.63625 


88 


2 


28.63625 


26.43160 


24.54176 


22.90377 


21.47040 


20.20555 


19.08114 


87 


3 


19.08114 


18.07498 


17.16934 


16.34986 


15.60478 


14.92442 


14.80067 


86 


4 


14.30067 


13.72674 


13.19688 


12.70621 


12.25051 


11.82617 


11.43005 


85 


5 


11.43005 


11.05943 


10.71191 


10.38540 


10.07803 


9.78817 


9.51436 


84 


6 


9.51436 


9.25530 


9.00983 


8.77689 


8.55555 


8.34496 


8.14435 


83 


7 


8.14435 


7.95302 


7.77035 


7.59575 


7.42871 


7.26873 


7.11537 


82 


8 


7.11537 


6.96823 


6.82694 


6.69110 


6.56055 


6.43484 


6.31375 


81 


9 


6.31375 


6.19703 


6.08444 


5.97576 


5.87080 


5.76937 


5.67128 


80 


10 


5.67128 


5.57638 


5.48451 


5.39552 


5.30928 


5.22566 


5.14455 


79 


11 


5.14455 


5.06584 


4.98940 


4.91516 


4.84300 


4.77286 


4.70463 


78 


12 


4.70463 


4.63825 


4.57363 


4.51071 


4.44942 


4.38969 


4.33148 


77 


13 


4.33148 


4.27471 


4.21933 


4.16530 


4.11256 


4.06107 


4.01078 


76 


14 


4.01078 


3.96165 


3.91364 


3.86671 


3.82083 


3.77595 


3.73205 


75 


15 


3.73205 


3.68909 


3.64705 


3.60588 


3. 5655 7 


3.52609 


3.48741 


74 


16 


3.48741 


3.44951 


3.41236 


3.37594 


3.34023 


3.30521 


3.27085 


73 


17 


3.27085 


3.23714 


3.20406 


3.17159 


3.13972 


3.10842 


3.07768 


72 


18 


3.07768 


3.04749 


3.01783 


2.98869 


2.96004 


2.93189 


2.90421 


71 


19 


2.90421 


2.87700 


2.85023 


2.82391 


2.79802 


2.77254 


2.74748 


70 


20 


2.74748 


2.72281 


2.69853 


2.67462 


2.65109 


2.62791 


2.60509 


69 


21 


.2.60509 


2.58261 


2.56040 


2.53865 


2.51715 


2.49597 


2.47509 


68 


22 


2.47509 


2.45451 


2.43422 


2.41421 


2.39449 


2.37504 


2.35585 


67 


23 


2.35585 


2.33693 


2.31826 


2.29984 


2.28167 


2.26374 


2.24604 


66 


24 


2.24604 


2.22857 


2.21132 


2.19430 


2.17749 


2.16090 


2.14451 


65 


25 


2.14451 


2.12832 


2.11233 


2.09654 


2.08094 


2.06553 


2.05030 


64 


26 


2.05030 


2.03526 


2.02039 


2.00569 


1.99116 


1.97680 


1.96261 


63 


27 


1.96261 


1.94858 


1.93470 


' 1.92098 


1.90741 


1.89400 


1.88073 


62 


28 


1.88073 


1.86760 


1.85462 


1.84177 


1.82907 


1.81649 


1.80405 


61 


29 


1.80405 


1.79174 


1.77955 


1.76749 


1.75556 


1.74375 


1.73205 


60 


30 


1.73205 


1.72047 


1.70901 


1.69760 


1.68643 


1.67530 


1.66428 


59 


31 


1.66428 


1.65337 


1.64256 


1.63185 


1.62125 


1.61074 


1.60033 


58 


32 


1.60033 


1.59002 


1.57981 


1.56969 


1.55966 


1.54972 


1.53987 


57 


33 


1.53987 


1.53010 


1.52043 


1.51084 


1.50133 


1.49190 


1.48256 


56 


34 


1.48256 


1.47330 


1.46411 


1.45501 


1.44598 


1.43703 


1.42815 


55 


35 


1.42815 


1 41934 


1.41061 


1.40195 


1.39336 


1.38484 


1.37638 


54 


36 


1.37638 


1.36800 


1.35968 


1.35142 


1.34323 


1.33511 


1.32704 


53 


37 


1.32704 


1.31904 


1.31110 


1.30323 


1.29541 


1.28764 


1.27994 


52 


38 


1.27994 


1.27230 


1.26471 


1.25717 


1.24969 


1.24227 


1.23490 


51 


39 


1.23490 


1.22758 


1.22031 


1.21310 


1.20593 


1.19882 


1.19175 


50 


40 


1.19175 


1.18474 


1.17777 


1.17085 


1.16398 


1.15715 


1.15037 


49 


41 


1.15037 


1.14363 


1.13694 


1.13029 


1.12369 


1.11713 


1.11061 


48 


42' 


1.11061 


1.10414 


1.09770 


1.09131 


1.08496 


1.07864 


1.07237 


47 


43 


1.07237 


1.06613 


1.05994 


1.05378 


1.04766 


1.04158 


1.03553 


46 


44 ' 


1.03553 


1.02952 


1.02355 


1.01761 


1.01170 


1.00583 


1.00000 


45 




60' 


60' 


40' 


30' 


20' 


10' 


0' 


I 










TANGENT 








& 



118 



20 
21 



24 

25 

26 
27 



30 
31 



1.00000 
1.0C015 
1.00061 
1.00137 
1.00244 

1.00382 
1.00551 
1.00751 
1.00983 
1.01247 

L. 01543 
1.01872 
1.02234 
1.02630 
1.03061 

1.03528 
1.04030 
1.04569 
1.05146 
1.05762 

1.06418 
1.07115 
1.07853 
1.08636 
1.09464 

1.10338 
1.112C0 
1.12233 
1.13257 
1.14335 

1.15470 
1.16663 
1.17918 
1 . 19236 



1.22077 
1.23607 
J52I4 
16902 
1.28676 

1.30541 
1.32501 
1.34563 
1.36733 
1.39016 



1.00001 
1.00021 
1.00072 
1.00153 
1.00265 

1.00408 
1.00582 
1 .00787 
1.01024 
1.01294 

1.01595 
1.00930 
1.02298 
1.02700 
1.03137 

1.03609 
1.04117 
1.04663 
1.05246 



1.06531 
1.07235 
1.07981 
1.08771 
1.09608 

1.10488 
1.11419 
1.12400 
1.13433 
1.14521 

1.15665 
1.16868 
1.18133 
1.19463 
1.20859 

1.22327 
1.23869 
1.25489 
1.27191 



1.32338 
1.34917 
1.37105 



1.00002 
1.00027 
1.00083 
1.0U1C9 
1.00287 

1.00435 
1.00614 
1.00325 
1.01067 
1.01342 

1.01049 
1.01989 
1.02362 
1.02770 
1.03213 

1.03691 
1.04206 
1.04757 
1.05347 
1.05976 

1.08645 
1.07356 
1.08109 
1.08907 
1.09750 

1.10640 
1.11579 
1.12568 
1.13610 
1.14707 

1.15861 
1.17075 
1.18350 
1.19691 
1.21099 

1.22579 
1.24134 
1.25767 
1.27483 
1.29287 

1.31183 
1.33177 
1.35274 
1 .37481 
1.39804 



1.00004 
1.00034 
1.00095 

1.00187 



1.00463 
1.00647 
1.00863 
1.01111 
1.01391 

1.01703 
1.02049 
1.02428 
1.02842 
1.03290 

1.03774 
1.04295 
1.04853 
1.05449 
1.06085 

1.06761 
1.07479 
1.08239 
1.09044 
1.09895 

1.10793 
1.11740 
1.12738 
1.13789 
1.14896 

1.16059 
1.17283 
1.18569 
1.19920 
1.21341 

1.22833 
1.24400 
1.26047 
1.27778 
1.29597. 

1.31509 
1.33519 
1.35634 
1.37860 
1.40203 



1.00007 I 1.00011 
1.00042 ! 1.00051 
1.00108 ; 1.00122 
1.00205 ! .1.00224 
1.00333 ! 1.00357 



1.00491 
1.00681 
1.00902 
1.01155 
1.01440 

1.01758 
1.02110 
1.02494 
1.02914 
1.03368 

1.03858 
1.04385 
1.04950 
1.05552 
1.06195 

1.06878 
1.07602 
1.0&370 
1.09183 
1.10041 

1.10947 
1.11903 
1.12910 
1 . 13970 
1.15085 

1.16259 
1.17493 
1.18?90 
1.20152 
1.21584 

1.23089 



1.28075 
1.29909 



1.35997 
1.38242 
1.40606 



40' | 30' 



1.00521 
1.00715 
1.00942 
1.01200 
1.01491 

1.01815 
] .02171 
1.02562 
1.02987 
1.03447 

1.03944 
1.04477 
1.05047 
1.05657 
1.06306 

1.06995 
1.07727 
1.08503 
1.0H323 
1.10189 

1.11103 
1.12067 
1.13083 
1.14152 
1.15277 

1.16460 
1.17704 
1.19012 
1.20386 
1.21830 

1.23347 
1.24940 
1.26615 
1.28374 
1.30223 



1.00015 
1.00061 
1.00137 
1.00244 

l.oor 

1.00551 
1.00751 
1.00983 
1.01247 
1.01543 

1.01872 



1.04 .. 
1.04569 
1.05146 
1.05762 
1.06418 

1.07115 
1.07853 
1.08636 
1.09464 
1.10 

1.11260 
1.12233 
1.13257 
1.14335 
1.15470 

1.16663 
1.17918 
1.19T " 



1.23607 
1.25214 



1.28676 
1.30541 



1.32501 
1.34563 



1.39016 
1.41421 



e 






COSECANTS 






IP 


0' 


10' 


20' 


30' 


40' 60' 


| 60' 







CO 


343.77516 


171.88831 


114.59301 


85.945611 68.75736 


57.29869 


89 


1 


57.29869 


49.11406 


42.97571 


38.20155 


34.38232 31.25758 


28.65371 


88 


2 


28.65371 


26.45051 


24.56212 


22.92559 


21.49368 '20.23028 


19.10732 


87 


3 


19.10732 


18.10262 


17.19843 


16.38041 


15.63679 14.95788 


14.33559 


86 


4 


.14.33559 


13.76312 


13.23472 


12.74550 


12.29125' 11.86837 


11.47371 


85 


5 


11.47371 


11.10455 


10.75849 


10.43343 


10.12752 9.83912 


9.56677 


84 


6 


9.56677 


9.30917 


9.06515 


8.83367 


8.61379 


8.46466 


8.20551 


83 


7 


8.20551 


8.01565 


7.83443 


7.66130 


7.49571 


7.33719 


7.18530 


82 


8 


7.18530 


7.03962 


6.89979 


6.76547 


6.63633 


6.51208 


6.39245 


81 


9 


6.39245 


6.27719 


6.16607 


6.05886 


5.95536 


5.85531 


5.75877 


80 


10 


5.75877 


5.66533 


5.57493 


5.48740 


5.40263' 5.32049 


5.24084 


79 


11 


5.24084 


5.16359 


5.08863 5.01585 


4.94517 


4.87649 


4.80973 


78 


12 


4.80973 


4.74482 


4.68167 


4.62023 


4.56041 


4.50216 


4.44541 


77 


13 


4.44541 


4.39012 


4.33622 


4.28366 


4.23239 


4.18238 


4.13357 


7fi 


14 


4.13357 


4.08591 


4.03938 


3.99393 


3.94952 


3.90613 


3.86370 


75 


15 


3.86370 


3.82223 


3.78166 


3.74198 


3.70315 


3.66515 


3.62796 


74 


.16 


3.62796 


3.59154 


3.55587 


3.52094 


3.48671 


3.45317 


3.42030 


73 


17 


3.42030 


3.38808 


3.35649 


3.32551 


3.29512 


3.26531 


3.23607 


72 


18 


3.23607 


3.20737 


3.17920 


3.15155 


3.12440 


3.09774 


3.07155 


71 


19 


3.07155 


3.04584 


3.02057 


2.99574 


2 97135 


.2.94737 


2.92380 


70 


20 


2.92380 


2.90063 


2.87785 


2.85545 


2.83342 


2.81175 


2.79043 


69 


21 


2.79043 


2.76945 


2.74881 


2.72850 


2.70851 


2.68884 


2.66947 


68 


22 


2.66947 


2.65040 


2.63162 


2.61313 


2.59491 


2.57698 


2.55930 


67 


23 


2.55930 


2.54190 


2.52474 


2.50784 


2.49119 


2.47477 


2.45859 


66 


.24 


2.45859 


2.44264 


2.42692 


2.41142 


2.39614 


2.38107 


2.36620 


65 


25 


2.36620 


2.35154 


2.33708 


2.32282 


2.30875 


2.29487 


2.28117 


64 


26 


2.28117 


2.26766 


2.25432 


2.24116 


2.22817 


2.21535 


2.20269 


63 


27 


2.20269 


2.19019 


2.17786 


2.16568 


2.15366 


2.14178 


2.13005 


62 


28 


2.13005 


2.11847 


2.10704 


2.09574 


2.08458 


2.07356 


2.06267 


61 


29 


2.06267 


2.05191 


2.04128 


2.03077 


2.02039 


2.01014 


2.00000 


60 


30 


2.00000 


1.98998 


1.98008 


1.97029 


1.96062 


1.95106 


1.94160 


59 


31 


1.94160 


1.93226 


1.92302 


1.91388 


1.90485 


1.89591 


1.88709 


58 


32 


1.88708 


1.87834 


1.86990 


1.86116 


1.85271 


1.84435 


1.83608 


57 


33 


1.83608 


1.82790 


1.81981 


1.81180 


1.80388 


1.79604 


1.78829 


56 


34 


1.78829 


1.78062 


1.77303 


1 76552 


1.75808 


1.75073 


1.74345 


55 


35 


1.74345 


1.73624 


1.72911 


1.72205 


1.71506 


1.70815 


1.70130 


54 


36 


1.70180 


1.69452 


1.68782 


1.68117 


1.67460 


1.66809 


1.66164 


53 


37 


1.66164 


1.65526 


1.64894 


1.64268 


1.63648 


1.63035 


1.62427 


52 


38 


1.62427 


1.61825 


1.61229 


1.60639 


1.60054 


1.59475 


1.58902 


51 


39 


1.58902 


1.58333 


1.57771 


1.57213 


1.56661 


1.56114 


1.55572 


50 


40 


1.55572 


1.55036 


1.54504 


1.53977 


1.53455 


1.52938 


1.52425 


49 


41 


1.52425 


1.51918 


1.51415 


1.50916 


1.50422 


1.49933 


1.49448 


48 


42 


1.49448 


1.48967 


1.48491 


1.48019 


1.47551 


1.47087 


1.46628 


47 


43 


1.46628 


1.46173 


1.45721 


1.45274 


1.44831 


1.44391 


1.43956 


40 


44 


1.43956 


1.43524 


1.43096 


1.42672 


1.42251 


1.41835 


1.41421 


45 




60' 


60' 


40' j 30' 


20' 


10' 


0' 


S 












1 




SECANTS 



120 



SQUARES, CUBES, SQUARE ROOTS, CUBE ROOTS, LOGARITHMS, RECIPROCALS, 
CIRCUMFERENCES AND CIRCULAR AREAS OF NOS. FROM 1 TO 1000 



No. 


Square 


Cube 


Sq. Root 


Cu. Root 


Log. 


lOOOxRecip. 


No. 


= Dia. 




















Circum. 


Area 


1 


1 


1 


1.0000 


1.0000 


0.00000 


1000.000 


3.142 


0.7854 


2 


4 


8 


1.4142 


1.2599 


0.30103 


500.000 


6.283 


3.1416 


3 


9 


27 


1.7321 


1.4422 


0.47712 


333.333 


9.425 


7.0686 


4 


16 


64 


2.0000 


1.5874 


0.60206 


250.000 


12.566 


12.5664 


5 


25 


125 


2.2361 


1.7100 


0.69897 


200.000 


15.708 


19.6350 


6 


36 


216 


2.4495 


1.8171 


0.77815 


166.667 


18.850 


28,2743 


7 


49 


343 


2.6458 


1.9129 


0.84510 


142.857 


21.991 


38.4845 


8 


64 


512 


2.8284 


2.0000 


0.90309 


125.000 


25.133 


50.2055 


9 


81 


729 


3.0000 


2.0801 


0.95424 


111.111 


28.274 


63.6173 


10 


100 


1000 


3.1623 


2.1544 


1.00000 


100.000 


31.416 


78.5398 


11 


121 


1331 


3.3166 


2.2240 


1.04139 


90.9091 


34.558 


95.0332 


12 


144 


1728 


3.4641 


2.2894 


1.07918 


83.3333 


37.699 


113.097 


13 


169 


2197 


3.6056 


2.3513 


1.11394 


76.9231 


40.841 


132.732 


14 


196 


2744 


3.7417 


2.4101 


1.14613 


71.4286 


43.982 


153.938 


15 


225 


3375 


3.8730 


2.4662 


1.17609 


66.6667 


47.124 


176.715 


16 


256 


4096 


4.0000 


2.5198 


1.20412 


62.5000 


50.265 


201.062 


17 


289 


4913 


4.1231 


2.5713 


1.23045 


58.8235 


53.407 


226.980 


18 


824 


5832 


4.2426 


2.6207 


1.25527 


55.5556 


56.549 


254.469 


19 


361 


6859 


4.3589 


2.6684 


1.27875 


52.6316 


59.690 


283.529 


20 


400 


8000 


4.4721 


2.7144 


1.30103 


50.0000 


62.832 


314.159 


21 


441 


9261 


4.5826 


2.7589 


1.32222 


47.6190 


65.973 


346.361 


82 


484 


10648 


4.6904 


2.8020 


1.34242 


45.4545 


69.115 


380.133 


23 


529 


12167 


4.7958 


2.8439 


1.36173 


43.4783 


72.257 


415.476 


24 


576 


13824 


4.8990 


2.8845 


1.38021 


41.6667 


75.398 


452.389 


25 


625 


15625 


5.0000 


2.9240 


1.39794 


40.0000 


78.540 


490.874 


26 


676 


17576 


5.0990 


2.9625 


1.41497 


38.4615 


81.681 


530.929 


27 


729 


19683 


5.1962 


3.0000 


1.43136 


37.0370 


84.823 


572.555 


28 


784 


21952 


5.2915 


3.0366 


1.44716 


35.7143 


87.965 


615.752 


29 


841 


24389 


5.3852 


3.0723 


1.46240 


34.4828 


91.106 


660.520 


BO 


900 


27000 


5.4772 


3.1072 


1.47712 


33.3333 


94.248 


706.858 


31 


961 


29791 


5.5678 


3.1414 


1.49136 


32.2581 


97.389 


754.768 


32 


1024 


32768 


5.6569 


3.1748 


1.50515 


31.2500 


100.531 


804.248 


88 


1089 


35937 


5.7446 


3.2075 


1.51851 


30.3030 


103.673 


855.299 


34 


1156 


39304 


5.8310 


3.2396 


1.53148 


29.4118 


106.814 


907.920 


85 


1225 


42875 


5.9161 


3.2711 


1.54407 


28.5714 


109.956 


962.113 


36 


1296 


46656 


6.0000 


3.3019 


1.55630 


27.7778 


113.097 


1017.88 


37 


1369 


50653 


6.0828 


3.3322 


1.56820 


27.0270 


116.239 


1075.21 


38 


1444 


54872 


6.1644 


3.3620 


1.57978 


26.3158 


119.381 


1134.11 


39 


1521 


59319 


6.2450 


3.3912 


1.59106 


25.6410 


122.522 


1194.59 


40 


1600 


64000 


6.3246 


3.4200 


1.60206 


25.0000 


125.66 


1256.64 


41 


1681 


68921 


6.4031 


3.4482 


1.61278 


24.3902 


128.81 


1320.25 


42 


1764 


74088 


6.4807 


3.4760 


1.62325 


23.8095 


131.95 


1385.44 


43 


1849 


79507 


6.5574 


3.5034 


1.63347 


23.2558 


135.09 


1452.20 


44 


1936 


85184 


6.6332 


3.5303 


1.64345 


22.7273 


138.23 


1520.53 


45 


2025 


91125 


6.7082 


3.5569 


1.65321 


22.2222 


141.37 


1590.43 


46 


2116 


97336 


0.7823 


3.5830' 


1.66276 


21.7391 


144.51 


1661.90 


47 


2209 


103823 


6.8557 


3.6088 


1.67210 


21.2766 


147.65 


1734.94 


48 


2804 


110592 


6.9282 


3.6342 


1.68124 


20.8333 


150.80 


1809.56 


49 


2401 


117649 


7.0000 


3.6593 


1.69020 


20.4082 


153.94 


1885.74 



121 



SQUARES, CUBES, SQUARE ROOTS, CUBE ROOTS, LOGARITHMS, RECIPROCALS, 
CIRCUMFERENCES AND CIRCULAR AREAS OF NOS.FROM 1 TO 1000 



No. 


Square 


Cube 


Sq. Root 


Cu. Root 


Log. 


lOOOxRecip 


No. = Dia. 










Circum. 


Area 


50 


2500 


125000 


7.0711 


3.6840 


1.69897 


20.0000 


157.08 


1963.50 


51 


2601 


132651 


7.1414 


3.7084 


1.70757 


19.6078 


160.22 


2042.82 


52 


2704 


140608 


7.2111 


3.7325 


1.71600 


19.2308 


163.36 


2123.72 


53 


2S09 


148877 


7.2801 


3.7563 


1.72428 


18.8679 


166.50 


2206.18 


54 


. 2916 


157464 


7.3485 


3.7798 


1.73239 


18.5185 


169.65 


2290.22 


55 


3025 


166375 


7.4162 


3.8030 


1.74036 


18.1818 


172.79 


2375.83 


56 


3136 


175616 


7.4833 


3.8259 


1.74819 


17.8571 


175.93 


2463.01 


57 


3249 


185193 


7.5498 


3.8485 


1.75587 


17.5439 


179.07 


2551.76 


58 


3364 


195112 


7.6158 


3.8709 


1.76343 


17.2414 


182.21 


2642.08 


59 


3481 


205379 


7.6811 


3.8930 


1.77085 


16.9492 


185.35 


2733.97 


60 


3600 


• 216000 


7.7460 


3.9149 


1.77815 


16.6667 


188.50 


2827.43 


61 


3721 


226981 


7.8102 


3.9365 


1.78533 


16.3934 


191.64 


2922.47 


62 


3844 


238328 


7.8740 


3.9579 


1.79239 


16.1290 


194.78 


3019.07 


63 


3969 


250047 


7.9373 


3.9791 


1.79934 


15.8730 


197.92 


3117.25 


64 


4096 


262144 


8.0000 


4.0000 


1.80618 


15.6250 


201.06 


3216.99 


65 


4225 


274625 


8.0623 


4.0207 


1.81291 


15.3846 


204.20 


3318.31 


66 


4356 


287496 


8.1240 


4.0412 


1.81954 


15.1515 


207.35 


8421.19 


67 


4489 


300763 


8.1854 


4.0615 


1.82607 


14.9254 


210.49 


3525.65 


68 


4624 


314432 


8.2462 


4.0817 


1.83251 


14.7059 


213.63 


3631.68 


69 


4761 


328509 


8.3066 


4.1016 


1.83885 


14.4928 


216.77 


3739.28 


70 


4900 


343000 


8.3666 


4.1213 


1.84510 


14.2857 


219.91 


3848.45 


71 


5041 


357911 


8.4261 


4.1408 


1.85126 


14.0845 


223.05 


3959.19 


72 


5184 


373248 


8.4853 


4.1602 


1.85733 


13.8889 


226.19 


4071.50 


73 


5329 


389017 


8.5440 


4.1793 


1.86332 


13.6986 


229.34 


4185.39 


74 


5476 


405224 


8.6023 


4.1983 


1.86923 


13.5135 


232.48 


4300.84 


75 


5625 


421875 


8.6603 


4.2172 


1.87506 


13.3333 


235.62 


4417.86 


76 


5776 


438978 


8.7178 


4.2358 


1.88081 


13.1579 


238.76 


4536.46 


77 


5929 


456533 


8.7750 


4.2543 


1.88649 


12.9870 


241. 90 


4656.63 


78 


6084 


474552 


8.8318 


4.2727 


1.89209 


12.8205 


245.04 


4778.36 


79 


6241 


493039 


8.8882 


4.2908 


1.89763 


12.6582 


248.19 


4901.67 


80 


6400 


512000 


8.9443 


4.3089 


1.90309 


12.5000 


251.33 


5026.55 


81 


6561 


531441 


9.0000 


4.3267 


1.90849 


12.3457 


254.47 


5153:00 


82 


6724 


551368. 


9.0554 


4.3445 


1.91381 


12.1951 


257.61 


5281.02 


83 


6889 


571787 


9.1104 


4.3621 


1.91908 


12.0482 


260.75 


5410.61 


84 


7056 


592704" 


9.1652 


4.3795 


1.92428 


11.9048 


263.89 


5541.77 


85 


7225 


614125 


9.2195 


4.3968 


1.92942 


11.7647 


267.04 


5674.50 


86 


7396 


636056 


9.2736 


4.4140 


1.93450 


11.6279 


270.18 


5808.80 


87 


7569 


658503 


9.3274 


4.4310 


1.93952 


11.4943 


273.32 


5944.68 


88 


7744 


681472 


9.3808 


4.4480 


1.94448 


11.3636 


276.46 


6082.12 


89 


7921 


704969 


9.4340 


4.4647 


1.94939 


11.2360 


279.60 


6221.14 


90 


8100 


729000 


9.4868 


4.4814 


1.95424 


11.1111 


282.74 


6361.73 


91 


8281 


753571 


9.5394 


4.4979 


1.95904 


10.9890 


285.88 


6503.88 


92 


8464 


778688 


9.5917 


4.5144 


1.96379 


10.8696 


289.03 


6647.61 


93 


8649 


804357 


9.6437 


4.5307 


1.96848 


10.7527 


292.17 


6792.91 


94 


8836 


830584 


9.6954 


4.5468 


1.97313 


10.6383 


295.81 


6939.78 


95 


9025 


857375 


9.7468 


4.5629 


1.97772 


10.5263 


298.45 


7088.22 


96 


9216 


884736 


9.7980 


4.5789 


1.98227 


10.4167 


301.59 


7238.23 


97 


9409 


912673 


9.8489 


4.5947 


1.98677 


10.3093 


304.73 


7389.81 


98 


9604 


941192 


9.8995 


4.6104 


1.99123 


10.2041 


307.88 


7542.96 


99 


9801 


970299 


9.9499 


4.6261 


1.99564 


10.1010 


311.02 


7697.69 



122 



SQUARES, CUBES, SQUARE ROOTS, CUBE ROOTS, LOGARITHMS, RECIPROCALS, 
CIRCUMFERENCES AND CIRCULAR AREAS OF NOS. FROM 1 T0 1000 



No. 


Square 


Cube 


Sq. Root 


Cm Root 


Log. 


lOOOxRecip. 


No. = Dia. 


















Circum. 


Area 


100 


10000 


1000000 


10.0000 


4.6416 


2.00000 


10.0000 


314.16 


7853.98 


101 


10201 


1030301 


10.0499 


4.6570 


2.00432 


9.90099 


317.30 


8011.85 


102 


10404 


1061208 


10.0995 


4.6723 


2.00860 


9.80392 


320.44 


8171.28 


103 


10609 


1092727 


10.1489 


4.6875 


2.01284 


9.70874 


323.58 


8332.29 


104 


10816 


1124864 


10.1980 


4.7027 


2.01703 


9.61538 


326.73 


8494.87 


105 


11025 


1157625 


10.2470 


4.7177 


2.02119 


9.52381 


329.87 


8659.01 


106 


11236 


1191016 


10.2956 


4.7326 


2.02531 


9.43396 


333.01 


8824.73 


107 


11449 


1225043 


10.3441 


4.7475 


2.02938 


9.34579 


336.15 


8992.02 


108 


11664 


1259712 


10.3923 


4.7622 


2.03342 


9.25926 


839.29 


9160.88 


109 


11881 


1295029 


10.4403 


4.7769 


2.03743 


9.17431 


342.43 


9331.32 


110 


12100 


1331000 


10.4881 


4.7914 


2.04139 


9.09091 


345.58 


9503.32 


111 


12321 


1367631 


10.5357 


4.8059 


2.04532 


9.00901 


348.72 


9676.89 


112 


12544 


1404928 


10.5830 


4.8203 


2.04922 


8.92857 


351.86 


9852.03 


113 


12769 


1442897 


10.6301 


4.8346 


2.05308 


8.84956 


355.00 


10028.7 


114 


12996 


1481544 


10.6771 


4.8488 


2.05690 


8.77193 


358.14 


10207.0 


115 


13225 


1520875 


10.7238 


4.8629 


2.06070 


8.69565 


361.28 


10386.9 


116 


13456 


1560896 


10.7703 


4.8770 


2.06446 


8.62069 


364.42 


10568.8 


117 


13689 


1601613 


10.8167 


4.8910 


2.06819 


8.54701 


367.57 


10751.3 


118 


13924 


1643032 


10.8628 


4.9049 


2.07188 


8.47458 


370.71 


10935; 9 


119 


14161 


1685159 


10.9087 


4.9187 


2.07555 


8.40336 


373.85 


11122.0 


120 


14400 


1728000 


10.9545 


4.9324 


2.07918 


8.33333 


376.99 


11309.7 


121 


14641 


1771561 


11.0000 


4.9461 


2.08279 


8.26446 


380.13 


11499.0 


122 


14884 


1815848 


11.0454 


4.9597 


2.08636 


8.19672 


383.27 


11689.9 


123 


15129 


1860867 


11.0905 


4.9732 


2.08991 


8.13008 


386.42 


11882.3 


124 


15376 


1906624 


11.1355 


4.9866 


2.09342 


8.06452 


889.56 


12076.3 


125 


15625 


1953125 


U.1803 
11.2250 


5.0000 


2.09691 


8.00000 


392.70 


12271.8 


126 


15876 


2000376 


5.0133 


2.10037 


7.93651 


395.84 


12469.0 


127 


16129 


2048383 


11.2694 


5.0265 


2.10380 


7.87402 


398.98 


12667.7 


12S 


16:384 


2097152 


11.3137 


5.0397 


2.10721 


7.81250 


402.12 


12868.0 


129 


16641 


2146689 


11.3578 


5.0528 


2.11059 


7.75194 


405.27 


13069.8 


130 


16900 


2197000 


11.4018 


5.0658 


2.11394 


7.69231 


408.41 


13273.2 


131 


17161 


2248091 


11.4455 


5.0788 


2.11727 


7.63359 


411.55 


13478.2 


132 


17424 


2299968 


11.4891 


5.0916 


2.12057 


7.57576 


414.69 


13684.8 


133 


17689 


2:352637 


11.5326 


5.1045 


2.12385 


7.51880 


417.83 


13892.9 


134 


17956 


2106104 


11.5758 


5.1172 


2.12710 


7.46269 


420.97 


14102.6 


133 


18225 


2460375 


11.6190 


5.1299 


2.13033 


7.40741 


424.12 


14313.9 


136 


18496 


2515456 


11.6619 


5.1426 


2,13354 


7.35294 


427.26 


14526.7 


137 


18769 


2571353 


11.7047 


5.1551 


2.13672 


7.29927 


430.40 


14741.1 


138 


19044 


2628072 


11.7473 


5.1676 


2.13988 


7.24638 


433.54 


14957.1 


139 


19321 


2685619 


11.7898 


5.1801 


2.14301 


7.19424 


436.68 


15174.7 


140 


19600 


2744000 


11.8322 


5.1925 


2.14613 


7.14286 


439.82 


15393.8 


141 


19881 


2803221 


11.8743 


5.2048 


2.14922 


7.09220 


442.96 


15614.5 


112 


20164 


2563288 


11.9164 


5.2171 


2.15229 


7.04255 


446.11 


15836.8 


143 


20449 


2924207 


11.9583 


5.2293 


2.15534 


6.99301 


449.25 


16060.6 


144 


20736 


2985984 


12.0000 


5.2415 


2.1583G 


6.94444 


452.39 


16286.0 


145 


21025 


3048625 


12.0416 


5.2536 


2.16137 


6.89655 


455.53 


16513.0 


140 


21316 


3112136 


12.0830 


5.2656 


2.16435 


6.84932 


458.67 


16741.5 


147 


21609 


3176523 


12.1244 


5.2776 


2.16732 


6.80272 


461.81 


16971.7 


148 


21904 


3241792 


12.1655 


5.2896 


2.17026 


6.75676 


464.96 


17203.4 


149 


22201 


3307949 


12.2066 


5.3015 


2.17319 


6.71141 


468.10 


17436.6 



123 



SQUARES, CUBES, SQUARE ROOTS, CUBE ROOTS, LOGARITHMS, RECIPROCALS, 
CIRCUMFERENCES AND CIRCULAR AREAS OF NOS. FROM 1 T0 1000 



No. 


Square 


Cube 


Sq. Root 


Cu. Root 


Log. 


lOOOxRecip. 


No. = Dia. 








Circum. 


Area 


150 


22500 


3375000 


12.2474 


5.3133 


2.17609 


6.66667 


471.24 


17671.5 


151 


22801 


3442951 


12.2882 


5.3251 


2.17898" 


6.62252 


474.38 


17907.9 


152 


23104 


3511808 


12.3288 


5.3368 


2.18184 


6.57895 


477.52 


18145.8 


153 


23409 


3581577 


12.3693 


5.3485 


2.18469 


6.53595 


480.66 


18385.4 


154 


23716 


3652264 


12.4097 


5.3601 


2.18752 


6.49351 


483.81 


18626.5 


155 


24025 


3723875 


12.4499 


5.3717 


2.19033 


6.45161 


486.95 


18809.2 


156 


24336 


3796416 


12.4900 


5.3832 


2.19312 


6.41026 


490.09 


19113.4 


157 


24649 


3869893 


12.5300 


5.3947 


2.19590 


6.36943 


493.23 


19359.3 


158 


24964 


3944312 


12.5698 


5.4061 


2.19866 


6.32911 


496.37 


19606.7 


159 


25281 


4019679 


12.6095 


5.4175 


2.20140 


6.28931 


499.51 


19855.7 


160 


25600 


4096000 


12.6491 


5.4288 


2.20412 


6.25000 


502.65 


20106.2 


161 


25921 


4173281 


12.6886 


5.4401 


2.20683 


6.21118 


505.80 


20358.3 


162 


26244 


4251528 


12.7279 


5.4514 


2.20952 


6.17284 


508.94 


20G12.0 


163 


26569 


4330747 


12.7671 


5.4626 


2.21219 


6.13497 


512.08 


20867.2 


164 


26896 


4410944 


12.8062 


5.4737 


2.21484 


6.09756 


515.22 


21124.1 


165 


27225 


4492125 


12.8452 


5.4848 


2.21748 


6.06061 


518.36 


21382.5 


166 


27556 


4574296 


12.8841 


5.4959 


2.22011 


6.02410 


521.50 


21642.4 


167 


27889 


4657463 


12.9228 


5.5069 


2.22272 


5.98802 


524.65 


21904.0 


168 


28224 


4741632 


12.9615 


5.5178 


2.22531 


5.95238 


527.79 


22167.1 


169 


28561 


4826809 


13.0000 


5.5288 


2.22789 


5.91716 


530.93 


22431.8 


170 


28900 


4913000 


13.0384 


5.5397 


2.23045 


5.88235 


534.07 


22698.0 


171 


29241 


5000211 


13.0767 


5.5505 


2.23300 


5.84795 


537.21 


22965.8 


172 


29584 


5088448 


13.1149 


5.5613 


2.23553 


5.81395 


540.35 


23235.2 


173 


29929 


5177717 


13.1529 


5.5721 


2.23805 


5.78035 


543.50 


23506.2 


174 


30276 


5268024 


13.1909 


5.5828 


2.24055 


5.74713 


•546.64 


23778.7 


175 


30625 


5359375 


13.2288 


5.5934 


2.24304 


5.71429 


549.78 


24052.8 


176 


30976 


5451776 


13.2665 


5.6041 


2.24551 


5.68182 


552.92 


24328.5 


177 


31329 


5545233 


13.3041 


5.6147 


2.24797 


5.64972 


556.06 


24605.7 


178 


31684 


5639752 


13.3417 


5.6252 


2.25042 


5.61798 


559.20 


24884.6 


179 


32041 


5735339 


13.3791 


5.6357 


2.25285 


5.58659 


562.35 


25164.9 


180 


32400 


5832000 


13.4164 


5.6462 


2.25527 


5.55556 


565.49 


25446.9 


181 


32761 


5929741 


13.4536 


5.6567 


2.25768 


5.52486 


568.63 


25730.4 


182 


33124 


6028568 


13.4907 


5.6671 


2.26007 


5.49451 


571.77 


26015.5 


183 


33489 


6128487 


13.5277 


5.6774 


2.26245 


5.46448 


574.91 


26302.2 


184 


33856 


6229504 


13.5647 


5.6877 


2.26482 


5.43478 


578.05 


26590.4 


185 


34225 


6331625 


13.6015 


5.6980 


2.26717 


5.40541 


581.19 


26880.3 


186 


34596 


6434856 


13.6382 


5.7083 


2.26951 


5.37634 


584.34 


27171.6 


187 


34969 


6539203 


13.6748 


5.7185 


2.27184 


5.34759 


587.48 


27464.6 


188 


35344 


6644672 


13.7113 


5.7287 


2.27416 


5.31915 


590.62 


27759.1 


189 


35721 


6751269 


13.7477 


5.7388 


2.27646 


5.29101 


593.76 


28055.2 


190 


36100 


6859000 


13.7840 


5.7489 


2.27875 


5.26316 


596.90 


28352.9 


191 


36481 


6967871 


13.8203 


5.7590 


2.28103 


5.23560 


600.04 


28652.1 


192 


36864 


7077888 


13.8564 


5.7690 


2.28330 


5.20833 


603.19 


2S952.9 


193 


37249 


7189057 


13.8924 


5.7790 


2.28556 


5.18135 


606.33 


29255.3 


194 


37636 


7301384 


13.9284 


5.7890 


2.28780 


5.15464 


609.47 


29559.2 


195 


38025 


7414875 


13.9642 


5.7989 


2.29003 


5.12821 


612.61 


29864. S 


196 


38416 


7529536 


14.0000 


5.8088 


2.29226 


5.10204 


615.75 


30171.9 


197 


38809 


7645373 


14.0357 


5.8186 


2.29447 


5.07614 


618.89 


30480.5 


198 


39204 


7762392 


14.0712 


5.8285 


2.29667 


5.05051 


622.04 


30790.7 


199 


39601 


7880599 


14.1067 


5.8383 


2.29885 


5.02513 


625.18 


3H02.6 



124 



SQUARES, CUBES, SQUARE ROOTS, CUBE ROOTS, LOGARITHMS, RECIPROCALS, 
CIRCUMFERENCES AND CIRCULAR AREAS OF NOS.FROM 1 T0 1000 



No. 


Square 1 


Cube 


Sq. Root 


Cu. Root 


Log. 


lOOOxRecip. 


No. = Dia. 




















Circum. 


Area 


200 


40000 


8000000 


14.1421 


5.8480 


2.30103 


5.00000 


628.32 


31415.9 


201 


40401 


8120601 


14.1774 


5.8578 


2.30320 


4.97512 


631.46 


31730.9 


.202 


40804 


8242408 


14.2127 


5.8675 


2.30535 


4.95050 


634.60 


32047.4 


203 


41209 


8365427 


14.2478 


5.8771 


2.30750 


4.92611 


637.74 


32365.5 


204 


41616 


8489664 


14.2829 


5.8868 


2.30963 


4.90196 


640.89 


32685.1- 


205 


42025 


8615125 


14.3178 


5.8964 


2.31175 


4.87805 


644.03 


33006.4 


206 


42436 


8741816 


14.3527 


5.9059 


2.31387 


4.85437 


647.17 


33329.2 


207 


42849 


8869743 


14.3875 


5.9155 


2.31597 


4.83092 


650.31 


33653.5 


208 


43264 


8998912 


14.4222 


5.9250 


2.31806 


4.80769 


653.45 


33979.5 


209 


43681 


9129329 


14.4568 


5.9345 


2.32015 


4.78469 


656.59 


34307.0 


210 


44100 


9261000 


14.4914 


5.9439 


2.32222 


4.76190 


659.73 


34636.1 


211 


44521 


9393931 


14.5258 


5.9533 


2.32428 


4.73934 


662.88 


34966.7 


212 


44944 


9528128 


14.5602 


5.9627 


2.32634 


4.71698 


666.02 


35298.9 


213 


45369 


9663597 


14.5945 


5.9721 


2.32838 


4.69484 


669.16 


35632.7 


214 


45796 


9800344 


14.6287 


5.9814 


2.33041 


4.67290 


672.30 


35968.1 


215 


46225 


9938375 


14.6629 


5.9907 


2.33244 


4.65116 


675.44 


36305. 


216 


46656 


10077696 


14.6969 


6.0000 


2.33445 


4.62963 


678.58 


36643.5 


217 


47089 


10218313 


14.7309 


6.0092 


2.33646 


4.60829 


681.73 


36983.6 


218 


47524 


10360232 


14.7648 


6.01&5 


2.33846 


4.58716 


684.87 


37325.3 


219 


47961 


10503459 


14.7986 


6.0277 


2.34044 


4.56621 


688.01 


37668.5 


220 


48400 


10648000 


14.8324 


6.0368 


2.34242 


4.54545 


691.15 


38013.3 


221 


48841 


10793861 


14.8661 


6.0459 


2.34439 


4.52489 


694.29 


38359.6 


222 


49284 


10941048 


14.8997 


6.0550 


2.34635 


4.50450 


697.43 


38707.6 


223 


49729 


11089567 


14.9332 


6.0641 


2.34830 


4.48431 


700.58 


39057,1 


224 


50176 


11239424 


14.9666 


6.0732 


2.35025 


4,46429 


703.72 


39408.1 


225 


50625 


11390625 


15.0000 


6.0822 


2.35218 


4.44444 


706.86 


39760.8 


226 


51076 


11543176 


15.0333 


6.0912 


2.35411 


4.42478 


710.00 


40115.0 


227 


51529 


11697083 


15.0665 


6.1002 


2.35603 


4.40529 


713.14 


40470.8 


228 


51984 


11852352 


15.0997 


6.1091 


2.35793 


4.38596 


716.28 


40828.1 


229 


52441 


12008989 


15.1327 


6.1180 


2.35984 


4.36681 


719.42 


41187.1 


230 


52900 


12167000 


15.1658 


6.1269 


2.36173 


4.34783 


722.57 


41547.6 


231 


53361 


12326391 


15.1987 


6.1358 


2.36361 


4.32900 


725.71 


41909.6 


232 


53824 


12487168 


15.2315 


6.1446 


2.36549 


4.31034 


728.85 


42273.3 


233 


54289 


12649337 


15.2643 


6.1534 


2.36736 


4.29185 


731.99 


42638.5 


234 


54756 


12812904 


15.2971 


6.1622 


2.36922 


4.27350 


735.13 


43005.3 


235 


55225 


12977875 


15.3297 


6.1710 


2.37107 


4.25532 


738.27 


43373.6 


236 


55696 


13144256 


15.3623 


6.1797 


2.37291 


4.23729 


741.42 


43T43.5 


237 


56169 


13312053 


15.3948 


6.1885 


2.37475 


4.21941 


744.56 


44115.0 


238 


56644 


13481272 


15.4272 


6.1972 


2.37658 


4.20168 


747.70 


44488.1 


239 


57121 


13651919 


15.4596 


6.2058 


2.37840 


4.18410 


750.84 


44862.7 


240 


57600 


13824000 


15.4919 


6.2145 


2.38021 


4.16667 


753.98 


45238.9 


241 


58081 


13997521 


15.5242 


6.2231 


2.38202 


4.14938 


757.12 


45616.7 


242 


58564 


14172488 


15.5563 


6.2317 


2.38382 


4.13223 


760.27 


45996.1 


243 


59049 


14348907 


15.5885 


6.2403 


2.38561 


4.11523 


763.41 


46377.0 


244 


59536 


14526784 


15.6205 


6.2488 


2.38739 


4.09836 


766.55 


46759.5 


245 


60025 


14706125 


15.6525 


6.2573 


2.38917 


4.08163 


769.69 


47143.5 


246 


60516 


14886936 


15.6844 


6.2658 


2.39094 


4.06504 


772.83 


47529.2 


247 


61009 


15069223 


15.7162 


6.2743 


2.39270 


4.04858 


775.97 


47916.4 


248 


61504 


15252992 


15.7480 


6.2828 


2.39445 


4.03226 


779.12 


48305.1 


249 


62001 


15438249 


15.7797 


6.2912 


2.39620 


4.01606 


782.26 


48695.5 



125 



SQUARES, CUBES, SQUARE ROOTS, CUBE ROOTS, LOGARITHMS, RECIPROCALS, 
CIRCUMFERENCES AND CIRCULAR AREAS OF NOS. FROM 1 TO 1000 



No. 


Square 


Cube 


Sq. Root 


Cu. Root 


Log. 


lOOOxRecip. 


No.= 


Dia. 








Circum. 


Area 


250 


62500 


15625000 


15.8114 


6.2996 


2.39794 


4.00000 


785.40 


49087.4 


251 


63001 


15813251 


15.8430 


6.3080 


2.39967 


3.98406 


788.54 


49480.9 


252 


63504 


16003008 


15.8745 


6.3164 


2.40140 


3.96825 


791.68 


49875.9 


253 


64009 


16194277 


15.9060 


6.3247 


2.40312 


3.95257 


794.82 


50272.6 


254 


64516 


16387064 


15.9374 


6.3330 


2.40483 


3.93701 


797.96 


50670.7 


255 


65025 


16581375 


15.9687 


6.3413 


2.40654 


3.92157 


801.11 


51070.5 


256 


65536 


16777216 


16.0000 


6.3496 


2.40824 


3.90625 


804.25 


51471.9 


257 


66049 


16974593 


16.0312 


6.3579 


2.40993 


3.89105 


807.39 


51874.8 


258 


66564 


17173512 


16 .0624 


6.3661 


2.41162 


3.87597 


810.53 


52279.2 


259 


67081 


17373979 


16.0935 


6.3743 


2.41330 


3.86100 


813.67 


52685.3 


260 


67600 


17576000 


16.1245 


6.3825 


2.41497 


3.84615 


816.81 


53092.9 


261 


68121 


17779581 


16.1555 


6.3907 


2.41664 


3.83142 


819.96 


53502.1 


262 


68644 


17984728 


16.1864 


6.3988 


2.41830 


3.81679 


823.10 


53912.9 


263 


69169 


18191447 


16.2173 


6.4070 


2.41996 


3.80228 


826.24 


54325.2 


264 


69696 


18399744 


16.2481 


6.4151 


2.42160 


3.78788 


829.38 


54739.1 


265 


70225 


18609625 


16.2788 


6.4232 


2.42325 


3.77358 


832.52 


55154.6 


266 


70756 


18821096 


16.3095 


6.4312 


2.42488 


3.75940 


835.66 


55571.6 


267 


71289 


19034163 


16.3401 


6.4393 


2.42651 


3.74532 


838.81 


55990.3 


268 


71824 


19248832 


16.3707 


6.4473 


2.42813 


3.73134 


841.95 


56410.4 


269 


72361 


19465109 


16.4012 


6.4553 


2.42975 


3.71747 


845.09 


56832.2 


270 


72900 


19683000 


16.4317 


6.4633 


2.43136 


3.70370 


848.23 


57255.5 


271 


73441 


19902511 


16.4621 


6.4713 


2.43297 


3.69004 


851.37 


57680.4 


272 


73984 


20123648 


16.4924 


6.4792 


2.43457 


3.67647 


854.51 


58106.9 


273 


74529 


20346417 


16.5227 


6.4872 


2.43616 


3.66300 


857.66 


58534.9 


274 


75076 


20570824 


16.5529 


6.4951 


2.43775 


3.64964 


860.80 


58964.6 


275 


75625 


20796875 


16.5831 


6.5030 


2.43933 


3.63636 


863.94 


59395.7 


276 


76176 


21024576 


16.6132 


6.5108 


2.44091 


3.62319 


867.08 


59828.5 


277 


76729 


21253933 


16.6433 


6.5187 


2.44248 


3.61011 


870.22 


60262.8 


278 


77284 


21484952 


16.6733 


6.5265 


2.44404 


3.59712 


873.36 


60698.7 


279 


77841 


21717639 


16.7033 


6.5343 


2.44560 


3.58423 


876.50 


61136.2 


280 


78400 


21962000 


16.7332 


6.5421 


2.44716 


3.57143 


879.65 


61575.2 


281 


78961 


22188041 


16.7631 


6.5499 


2.44871 


3.55872 


882.79 


62015.8 


282 


79524 


22425768 


16.7929 


6.5577 


2.45025 


3.54610 


885.93 


62458.0 


283 


80089 


22665187 


16.8226 


6.5654 


2.45179 


3.53357 


889.07 


62901.8 


284 


80656 


22906304 


16.8523 


6.5731 


2.45332 


3.52113 


892.21 


63347.1 


285 


81225 


23149125 


16.8819 


6.5808 


2.45484 


3.50877 


895.35 


63794.0 


286 


81796 


23393656 


16.9115 


6.5885 


2.45637 


3.49650 


898.50 


64242.4 


287 


82369 


23639903 


16.9411 


6.5962 


2.45788 


3.48432 


901.64 


64692.5 


288 


82944 


23887872 


16.9706 


6.6039 


2.45939 


3.47222 


904.78 


65144.1 


289 


83521 


24137569 


17.0000 


6.6115 


2.46090 


3.46021 


907.92 


65597.2 


290 


84100 


24389000 


17.0294 


6.6191 


2.46240 


3.44828 


911.06 


66052.0 


291 


84681 


24642171 


17.0587 


6.6267 


2.46389 


3.43643 


914.20 


66508.3 


292 


85264 


24897088 


. 17.0880 


6.6343 


2.46538 


3.42466 


917.35 


66966.2 


293 


85849 


25153757 


17.1172 


6.6419 


2.46687 


3.41297 


: 920.49 


67425.6 


294 


86436 


25412184 


17.1464 


6.6494 


. 2.46835 


3.40136 


923.63 


67886.7 


295 


87025 


25672375 


17.1756 


6.6569 


2.46982 


3.38983 


926.77 


68349.3 


296 


87616 


25934336 


17.2047 


6.6644 


2.47129 


3.37838.. 


929.91 


68813.5 


297 


88209 


26198073 


17.2337 


6.6719 


2.47276 


3.36700 


933.05 


69279.2 


298 


88804 


26463592 


17.2627 


6.6794 


2.47422 


3.35570 


936.19 


69746.5 


299 


89401 


26730899 


17.2910 


6.6869 


2.47567 


3.34448 


939.34 


70215.4 



126 



SQUARES, CUBES, SQUARE ROOTS, CUBE ROOTS, LOGARITHMS, RECIPROCALS, 
CIRCUMFERENCES AND CIRCULAR AREAS OF NOS. FROM 1 TO 1000 



No. 


Square 


Cube 


Sq. Root 


Cu. Root 


Log. 


lOOOxRecip. 


No. = Dia. 




















Circum. 


Area 


300 


90000 


27000000 


17.3205 


6.6943 


2.47712- 


3.33333 


942.48 


70685.8 


301 


90601 


27270901 


17.3494 


6.7018 


2.47857 


3.32226 


945.62 


71157.9 


302 


91204 


27543608 


17.3781 


6.7092 


2.48001 


3.31126 


948.76 


71631.5 


303 


91809 


27818127 


17.4069 


6.7166 


2.48144 


3.30033 


951.90 


72106.6 


304 


92416 


28094464 


17.4356 


6.7240 


2.48287 


3.28947 


955.04 


72583.4 


305 


93025 


28372625 


17.4642 


6.7313 


2.48430 


3.27869 


958.19 


73061.7 


306 


93636 


28652616 


17.4929 


6.7387 


2.48572 


3.26797 


961.33 


78541.5 


307 


94249 


28934443 


17.5214 


6.7460 


2.48714 


8.25733 


964.47 


74023.0 


308 


94864 


29218112 


17.5499 


6.7533 


2.48855 


3.24675 


967.61 


74506.0 


309 


95481 


29503629 


17.5784 


6.7606 


2.48996 


3.23625 


970.75 


74990.6 


310 


96100 


29791000 


17.6068 


6.7679 


2.49136 


3.22581 


973.89 


75476.8 


311 


96721 


30080231 


17.6352 


6.7752 


2.49276 


3.21543 


977.04 


75964.5 


312 


97344 


30371328 


17.6635 


6.7824 


2.49415 


3.20513 


980.18 


76453.8 


313 


97969 


30664297 


17.6918 


6.7897 


2.49554 


3.19489 


983.32 


76944.7 


314 


98596 


30959144 


17.7200 


6.7969 


2.49693 


3.18471 


986.46 


77437.1 


315 


99225 


31255875 


17.7482 


6.8041 


2.49831 


3.17460 


989.60 


77931.1 


316 


99856 


31554496 


17.7764 


6.8113 


2.49969 


3.16456 


992.74 


78426.7 


317 


100489 


31855013 


17.8045 


6.8185 


2.50106 


3.15457 


995.88 


78923.9 


318 


101124 


32157432 


17.8326 


6.8256 


2.50243 


3.14465 


999.03 


79422 6 


319 


101761 


32461759 


17.8606 


6.8328 


2.50379 


3.13480 


1002.2 


79922.9 


320 


102400 


32768000 


17.8885 


6.8399 


2.50515 


3.12500 


1005.3 


80424.8 


321 


103041 


33076161 


17.9165 


6.8470 


2.50651 


3.11527 


1008.5 


80928.2 


322 


103684 


33386248 


17.9444 


6.8541 


2.50786 


3.10559 


1011.6 


81433.2 


323 


104329 


33698267 


17.9722 


6.8612 


2.50920 


3.09598 


1014.7 


81939.8 


324 


104976 


34012224 


18.0000 


6.8683 


2.51055 


3.08642 


1017.9 


82448.0 


325 


105625 


34328125 


18.0278 


6.8753 


2.51188 


8.07692 


1021.0 


82957.7 


326 


106276 


34645976 


18.0555 


6.8824 


2.51322 


3.06749 


1024.2 


83469.0 


327 


106929 


34965783 


18.0831 


6.8894 


2:51455 


8.05810 


1027.3 


83981.8 


328 


107584 


35287552 


18.1108 


6.8964 


2.51587 


3.04878 


1030.4 


84496.3 


329 


108241 


35611289 


18.1384 


6.9034 


2.51720 


3.03951 


1033.6 


85012.3 


330 


108900 


35937000 


18.1659 


6.9104 


2.51851 


3.0S030 


1036.7 


85529.9 


331 


109561 


36264691 


18.1934 


6.9174 


2.51983 


3.02115 


1039.9 


86049.0 


332 


110224 


36594368 


18.2209 


6.9244 


2.52114 


3.01205 


1043.0 


86569.7 


333 


110889 


36926037 


18.2483 


6.9313 


2.52244 


3.00300 


1046.2 


87092.0 


384 


111556 


37259704 


18.2757 


6.9382 


2.52375 


2.99401 


1049.3 


87615.9 


335 


112225 


37595375 


18.3030 


6.9451 


2.52504 


2.98507 


1052.4 


88141.3 


336 


112896 


37933056 


18.3303 


6.9521 


2.52634 


2.97619 


1055.6 


88668 .3 


337 


113569 


38272753 


18.3576 


6.9589 


2.52763 


2.96736 


1058.7 


89196.9 


338 


114244 


38614472 


18.3848 


6.9658 


2.52892 


2.95858 


1061.9 


89727.0 


339 


114921 


38958219 


18.4120 


6.9727 


2.53020 


2.94985 


1065.0 


90258.7 


340 


115600 


39304000 


18.4391 


6.9795 


2.53148 


2.94118 


1068.1 


90792.0 


841 


116281 


S9651821 


18.1662 


6.9864 


2.53275 


2.93255 


1071.3 


91826.9 


342 


116964 


40001688 


18.4932 


6.9932 


2.53403 


2.92398 


1074.4 


91868.3 


343 


117649 


40353607 


18.5203 


7.0000 


2.53529 


2.91545 


1077.6 


92401.3 


344 


118336 


40707584 


18.5472 


7.0068 


2.53656 


2.90698 


1080'. 7 


92940.9 


345 


119025 


41063625 


18.5742 


7.0136 


2.53782 


2.89855 


1083.8 


93482.0 


34ii 


119716 


41421736 


18.6011 


7.0203 


2.53908 


2.89017 


1087.0 


94024.7 


347 
348 


120409 


41781923 


18.6279 


7.0271 


2.54033 


2.88184 


1090.1 


94569.0 


121104 


42144192 


18.6548 


7.0338* 


2.54158 


2.87356 


1093.3 


95114.9 


319 


121801 


42508549 


18.6815 


7.0406 


2.54283 


2.86533 


1096.4 


95662.3 



127 



SQUARES, CUBES, SQUARE ROOTS, CUBE ROOTS, LOGARITHMS, RECIPROCALS, 
CIRCUMFERENCES AND CIRCULAR AREAS OF NOS. FROM 1 T0 1000 



Ho. 


Square 


Cube 


Sq. Root 


Cu. Root 


log. 


iOOOxRecip. 


Ho. = 


Dia. 






















Circum. 


Area 


350 


122500 


42875000 


18.7083 


7.0473 


2.54407 


2.85714 


1099.6 


96211.3 


351 


123201 


43243551 


18.7350 


7.0540 


2.54531 


2.84900 


1102.7 


96761.8 


352 


123904 


43614208 


18.7617 


7.0607 


2.54654 


2.84091 


1105.8 


97314.0 


353 


124609 


43986977 


18.7883 


7.0674 


2.54777 


2.83286 


1109.0 


97867.7 


354 


125316 


44361864 


18.8149 


7.0740 


2.54900 


2.82486 


1112.1 


98423.0 


355 


126025 


44738875 


18.8414 


7.0807 


2.55023 


2.81690 


1115.3 


98979.8 


356 


126736 


45118016 


18.8680 


7.0873 


2.55145 


2.80899 


1118.4 


99538.2 


357 


127449 


45499293 


18.8944 


7.0940 


2.55267 


2.80112 


1121,5 


100098 


358 


128164 


45882712 


18.9209 


7.1006 


2.55388 


2.79330 


1124.7 


100660 


359 


128881 


46268279 


18.9473 


7.1072 


2.55509 


2.78552 


1127.8 


101223 


360 


129600 


46656000 


18.9737 


7.1138 


2.55630 


2.77778 


1131.0 


101788 


361 


130321 


47045881 


19.0000 


7.1204 


2.55751 


2.77008 


1134.1 


102354 


362 


131044 


47437928 


19.0263 


7.1269 


2.55871 


2.76243 


1137.3 


102922 


363 


131769 


47832147 


19.0526 


7.1335 


2.55991 


2.75482 


1140.4 


103491 


364 


132496 


48228544 


19.0788 


7.1400 


2.56110 


2.74725 


1143.5 


104062 


365 


133225 


48627125 


19.1050 


7.1466 


2.56229 


2.73973 


1146.7 


104635 


366 


133956 


49027896 


19.1311 


7.1531 


2.56348 


2.73224 


-1149.8 


105209 


367 


134689 


49430863 


19.1572 


7.1596 


2.56467 


2.72480 


1153.0 


105785 


368 


135424 


49836032 


19.1833 


7.1661 


2.56585 


2.71739 


1156.1 


106362 


369 


136161 


50243409 


19.2094 


7.1726 


2.56703 


2.71003 


1159.2 


106941 


370 


136900 


50653000 


19.2354 
19.2614 


7.1791 


2.56820 


2.70270 


1162.4 


107521 


371 


137641 


51064811 


7.1855 


2.56937 


2.69542 


1165.5 


108103 


372 


138384 


51478848 


19.2873 


7.1920 


2.57054 


2.68817 


1168.7 


108687 


373 


139129 


51895117 


19.3132 


7.1984 


2.57171 


2.68097 


1171.8 


109272 


374 


139876 


52313624 


19.3391 


7.2048 


2.57287 


2.67380 


1175.0 


109858 


375 


140625 


52734375 


19.3649 


Z. 2112 


2.57403 


2.66667 


1178.1 


110447 


376 


141376 


53157376 


19.3907 


7.2177 


2.57519 


2.65957 


1181.2 


111036 


377 


142129 


53582633 


19.4165 


7.2240 


2.57634 


'2.65252 


1184.4 


111628 


378 


142884 


54010152 


19.4422 


7.2304 


2.57749 


2.64550 


1187.5 


112221 


379 


143641 


54439939 


19.4679 


7.2368 


2,57864 


2.63852 


1190.7 


112815 


380 


144400 


54872000 


19.4936 


7.2432 


2.57978 


2.63158 


1193.8 


113411 


381 


145161 


55306341 


19.5192 


7.2495 


2.58093 


2.62467 


1196.9 


114009 


382 


145924 


55742968 


19.5448 


7.2558 


2.58206 


2.61780 


1200.1 


114608 


383 


146689 


56181887 


19.5704 


7.2622 


2.58320 


2.61097 


1203.2 


115209 


384 


147456 


56623104 


19.5959 


7.2685 


2.58433 


2.60417 


1206.4 


115812 


385 


148225 


57066625 


19.6214 


7.2748 


2.58546 


2.59740 


1209.5 


116416 


386 


148996 


57512456 


19.6469 


7.2811 


2.58659 


2.59067 


1212.7 


117021 


387 


149769 


57960603 


19.6723 


7.2874 


2.58771 


2.58398 


1215.8 


117628 


388 


150544 


58411072 


19.6977 


7.2936 


2.58883 


2.57732 


1218.9 


118237 


389 


151321 


58863869 


19.7231 


7.2999 


2.58995 


2.57069 


1221.1 


118847 


390 


152100 


59319000 


19.7484 


7.3061 


2.59106 


2.56410 


1225.2 


119459 


391 


152881 


59776471 


19.7737 


7.3124 


2.59218 


2.55755 


1228.4 


120072 


392 


153664 


60236288 


19.7990 


7.3186 


2.59329 


2.55102 


1231.5 


120687 


393 


154449 


60698457 


19.8242 


7.3248 


2.59439 


2.54453 


1234.6 


121304 


394 


155236 


61162984 


19.8494 


7.3310 


2.59550 


2.53807 


1237.8 


121922 


395 


156025 


61629875 


19.8746 


7.3372 


2.59660 


2.53165 


1240.9 


122542 


396 


156816 


62099136 


19.8997 


7.3434 


2.59770 


2.52525 


1244.1 


123163 


397 


157609 


62570773 


19.9249 


7.3496 


2.59879 


2.51889 


1247.2 


123786 


398 


158404 


63044792 


19.9499 


7.3558 


2.59988 


2.51256 


1250.4 


124410 


399 


159201 


63521199 


19.9750 


7.3619 


2.60097 


2.50627 


1253.5 


125036 



128 



SQUARES, CUBES, SQUARE ROOTS, CUBE ROOTS, LOGARITHMS, RECIPROCALS, 
CIRCUMFERENCES AND CIRCULAR AREAS OF NOS. FROM 1 T0 1000 



Ho. 


Square 


Cube 


Sq. Root 


Cu. Root 


Log. 


lOOOiRecip 


No. = Dia. 










Circum. 


Area 


400 


160000 


64000000 


20.0000 


7.3681 


2.60206 


2.50000 


1256.6 


125664 


401 


160801 


64481201 


20.0250 


7.3742 


2.60314 


2.49377 


1259.8 


126293 


402 


161604 


64964808 


20.0499 


7.3803 


2.60423 


2.48756 


1262.9 


126923 


403 


162409 


65450827 


20.0749 


7.3864 


2.60531 


2.48139 


1266.1 


127556 


404 


163216 


65939264 


20.0998 


7.3925 


2.60638 


2.47525 


1269.2 


128190 


405 


164025 


66430125 


20.1246 


7.3986 


2.60746 


2.46914 


1272.3 


128825 


406 


164836 


66923416 


20.1494 


7.4047 


2.60853 


2.46305 


1275.5 


129462 


407 


165649 


67419143 


20.1742 


7.4108 


2.60959 


2.45700 


1278.6 


130100 


408 


166464 


67917312 


20.1990 


7.4169 


2.61066 


2.45098 


1281.8 


130741 


409 


167281 


68417929 


20.2237 


7.4229 


2.61172 


2.44499 


1284.9 


131382 


410 


168100 


68921000 


20.2485 


7.4290 


2.61278 


2.43902 


1288.1 


132025 


411 


168921 


69426531 


20.2731 


7.4350 


2.61384 


2.43309 


1291.2 


132670 


412 


169744 


69934528 


20.2978 


7.4410 


2.61490 


2.42718 


1294.3 


133317 


413 


170569 


70444997 


20.3224. 


7.4470 


2.61595 


2.42131 


1297.5 


133965 


414 


171396 


70957944 


20.3470 


7.4530 


2.61700 


2.41546 


1300.6 


134614 


415 


172225 


71473375 


20.3715 


7.4590 


2.61805 


2.40964 


1303.8 


135265 


416 


173056 


71991296 


20.3961 


7.4650 


2.61909 


2.40385 


1306.9 


135918 


417 


173889 


72511713 


20.4206 


7.4710 


2.62014 


2.39808 


1310.0 


136572 


418 


174724 


73034632 


20.4450 


7.4770 


2.62118 


2.39234 


1313.2 


137228 


419 


175561 


73560059 


20.4695 


7.4829 


2.62221 


2.38664 


1316.3 


137885 


420 


176400 


74088000 


20.4939 


7.4889 


2.62325 


2.38095 


1819.5 


138544 


421 


177241 


74618461 


20.5183 


7.4948 


2.62428 


2.37530 


1322.6 


139205 


422 


178C84 


75151448 


20.5426 


7.5007 


2.62531 


2.36967 


1325.8 


139867 


423 


178929 


75686967 


20.5670 


7.5067 


2.62634 


2.36407 


1328.9 


140531 


424 


179776 


76225024. 


20.5913 


7 5126 


2.62737 


2.35849 


1332.0 


141196 


425 


1K0625 


76765625 


20.6155 


7.5185 


2.62839 


2.35294 


1335.2 


141863 


426 


181476 


77308776 


20.6398 


7.5244 


2.62941 


2.34742 


1338.3 


142531 


427 


182329 


77854483 


20.6640 


7.5302 


2.63043 


2.34192 


1341.5 


143201 


428 


183184 


78402752 


20.6882 


7.5361 


2.63144 


2.33645 


1344.6 


143872 


429 


184041 


78953589 


20.7123 


7.5420 


2.63246 


2.33100 


1347.7 


144545 


430 


184900 


79507000 


20.7364 


7.5478 


2.63347 


2.32558 


1350.9 


145220 


431 


185761 


80062991 


20.7605 


7.5537 


2.63448 


2.32019 


1354.0 


145896 


432 


186624 


80621568 


20.7846 


7.5595 


2.63548 


2.31482 


1357.2 


146574 


433 


187489 


81182737 


20.8087 


7.5654 


2.63649 


2.30947 


1360.3 


147254 


434 


188356 


81746504 


20.8327 


7.5712 


2.63749 


2.30415 


1363.5 


147934 


435 


189225 


82312875 


20.8567 


7.5770 


2.63849 


2.29885 


1366.6 


148617 


436 


190096 


82881856 


20.8806 


7.5828 


2.63949 


2.29358 


1369.7 


149301 


437 


190969 


83453453 


20.9045 


7.5886 


2.64048 


2.28833 


1372.9 


149987 


438 


191844 


84027672 


20.9284 


7.5944 


2.64147 


2.28311 


1376.0 


150674 


439 


192721 


84604519 


20.9523 


7.6001 


2.64246 


2.27790 


1379.2 


151363 


440 


193600 


85184000 


20 9762 


7 6059 


2.64345 


2.27273 


1382.3 


152053 


441 


194481 


85766121 


21.0000 


7.6117 


2.64444 


2.26757 


1385.4 


152745 


442 


195364 


86350888 


21.0238 


7.6174 


2.64542 


2.26244 


1388.6 


153439 


443 


196249 


86938307 


21.0476 


7.6232 


2.64640 


2.25734 


1391.7 


154134 


444 


197136 


87528384 


21.0713 


7.6289 


2.64^38 


2.25225 


1394.9 


154830 


445 


198025 


88121125 


21.0950 


7.6346 


2.64836 


2.24719 


1398.0 


155528 


446 


198916 


88716536 


21.1187 


7.6403 


2.64933 


2.24215 


1401.2 


156228 


447 


199809 


89314623 


21.1424 


7.6460 


2.65031 


2.23714 


1404.3 


156930 


448 


200704 


89915392 


21.1660 


7.6517 


2.65i28 


2.23214 


1407.4 


157633 


449 


201601 


90518849 


21.1896 


7.6574 


2.65225 


2.22717 


1410.6 


158337 



129 



SQUARES, CUBES, SQUARE ROOTS, CUBE ROOTS, LOGARITHMS, RECIPROCALS, 
CIRCUMFERENCES AND CIRCULAR AREAS OF NOS. FROM 1 TO 1000 



No. 


Square 


Cube Sq.Root 


Cu. Root. 


Log. 


lOOOxRecip 


No. — Dia. 








Circum. 


Area 


450 


202500 


91125000 


21.2132 


7.6631 


2.65321 


2.22222 


1413.7 


159043 


451 


203401 


91733851 


21.2368 


7.6688 


2.65418 


2.21730 


1416.9 


159751 


452 


204304 


92345108 


21.2603 


7.6744 


2.65514 


2.21239 


1420.0 


160460 


453 


205209 


92959677 


21.2838 


7.6801 


2.65610 


2.20751 


1423.1 


161171 


454 


206116 


93576664 


21.3073 


7.6857 


2.65706 


2.20264 


1426.3 


161883 


455 


207025 


94196375 


21.3307 


7.6914 


2.65801 


2.19780 


1429.4 


162597 


456 


207936 


94818816 


21.3542 


7.6970 


2.65896 


2.19298 


1432.6 


163313 


457 


208849 


95443993 


21.3776 


7.7026 


2.65992 


2.18818 


1435.7 


164030 


458 


209764 


96071912 


21.4009 


7.7082 


2.66087 


2.18341 


1438.9 


164748 


459 


210681 


96702579 


21.4243 


7.7138 


2.66181 


2.17865 


1442.0 


165468 


460 


211600 


9733G000 


21.4476 


7.7194 


2.66276 


2.17391 


1445.1 


166190 


461 


212521 


97972181 


21.4709 


7.7250 


2.66370 


2.16920 


1448.3 


166914 


462 


213444 


98611128 


21.4942 


7.7306 


2.66464 


2.16450 


1451.4 


167639 


463 


214369 


99252847 


21.5174 


7.7362 


2.66558 


2.15983 


1454.6 


168365 


464 


215296 


99897344 


21.5407 


7.7418 


2.66652 


2.15517 


1457.7 


169093 


465 


216225 


100544625 


21.5639 


7.7473 


2.66745 


2.15054 


1460.8 


169823 


466 


217156 


101194696 


21.5870 


7.7529 


2.66839 


2.14592 


1464.0 


170554 


467 


218089 


101847563 


21.6102 


7.7584 


2.66932 


2.14133 


1467.1 


171287 


468 


210024 


102503232 


21.6333 


7.7639 


2.67025 


2.13675 


1470.3 


172021 


469 


219961 


103161709 


21.6564 


7.7695 


2.67117 


2.13220 


1473.4 


172757 


470 


220900 


103823000 


21.6795 


7.7750 


2.67210 


2.12766 


1476.5 


173494 


471 


221841 


104487111 


21.7025 


7.7805 


2.67302 


2.12314 


1479.7 


174234 


472 


2227N1 


105154048 


21.7256 


7.7860 


2.67394 


2.11864 


1482.8 


174974 


473 


223729 


105823817 


21.7486 


7.7915 


2.67486 


2.11417 


1486.0 


175716 


474 


224676 


106496424 


21.7715 


7.7970 


2.67578 


2.10971 


1489.1 


176460 


475 


225(125 


107171875 


21.7945 


7.8025 


2.67669 


2.10526 


1492.3 


177205 


476 


226576 


107850176 


21.8174 


7.8079 


2.67761 


2.10084 


1495.4 


177952 


477 


227520 


108531333 


21.8403 


7.8134 


2.67852 


2.09644 


1498.5 


178701 


478 


228484 


109215352 


21.8632 


7.8188 


2.67943 


2.09205 


1501.7 


179451 


479 


229441 


109902239 


21.8861 


7.8243 


2.68034 


2.08768 


1504.8 


180203 


480 


230400 


110592000 


21 .9089 


7.8297 


2.68124 


2.08333 


1508.0 


180956 


481 


231361 


111284641 


21.9317 


7.8352 


2.68215 


2.07900 


1511.1 


181711 


482 


232324 


111980168 


21.9545 


7.8406 


2.68305 


2.07469 


1514.3 


182467 


483 


233289 


112678587 


21.9773 


7.8460 


2.68395 


2.07039 


1517.4 


!S322f> 


484 


234256 


113379904 


22.0000 


7.8514 


2.68485 


2.06612 


1520.5 


183984 


485 


235225 


114084125 


22.0227 


7.8568 


2.68574 


2.06186 


1523.7 


184745 


486 


236196 


114791256 


22.0454 


7.8622 


2.68664 


2.05761 


1526.8 


185508 


487 


237169 


115501303 


22.0681 


7.8676 


2.68753 


2.05339 


1530.0 


-186272 


488 


238144 


116214272 


22.0907 


7.-8730 


2.68842 


2.04918 


1533.1 1 187038 


489 


239121 . 


116930169 


22.1133 


7.8784 


2.68931 


2.04499 


1536.2 ; 187805 


490 


240100 


117649000 


22.1359 


7,8837 


2.69020 


2.04082 


1539.4 


188574 


491 


241081 


118370771 


22.1585 


7.8891 


2.69108 


2.03665 


1542.5 


189345 


492 


242064 


119095488 


22.1811 


7.8944 


2.69197 


2.03252 


1545.7 


190117 


493 


243049 


119823157 


22.2036 


7.8998 


2.69285 


2.02840 


1548.8 


190890 


494 


244036 


120553784 


22.2261 


7.9051 


2.69373 


2.02429 


1551.9 


191665 


495 


245025 


121287375 


22.2486 


7.9105 


2.69461 


2.02020 


1555.1 


192442 


496 


246016 


122023936 


22.2711 


7.9158 


2.69548 


2.01613 


1558.2 


193221 


497 


247009 


122763473 


22.2935 


7.9211 


2.69636 


2.01207 


1561.4 


194000 


498 


248004 


123505992 


22.3159 


7.9264 


2.69723 


2.00803 


1564.5 


194782 


499 


249001 


124251499 


22.3383 


7.9317 


2.69810 


2.00401 


1567.7 


195565 



130 



SQUARES, CUBES, SQUARE ROOTS, CUBE ROOTS, LOGARITHMS, RECIPROCALS. 
CIRCUMFERENCES AND CIRCULAR AREAS OF NOS. FROM 1 TO 1000 



No. 


Square 


Cube 


Sq. Root 


Cu. Root 


Log. 


lOOOxRecip. 


No. = 
Circum. 


Dia. 
Area 


500 


250000 


125000000 


22.3607 


7.9370 


2.69897 


2.00000 


1570.8 


196350 


501 


251001 


125751501 


22.3830 


7.9423 


2.69984 


1.99601 


1573.9 


197136 


502 


252004 


126506008 


22.4054 


7.9476 


2.70070 


1.99203 


1577.1 


197923 


503 


253009 


127203527 


22.4277 


7.9528 


2.70157 


1.98807 


1580.2 


198713 


504 


254016 


128024064 


22.4499 


7.9581 


2.70243 


1.98413 


1583.4 


199504 


505 


255025 


128787625 


22.4722 


7.9634 


2.70329 


1.98020 


1586.5 


200296 


506 


256036 


129554216 


22.4944 


7.9686 


2.70415 


1.97629 


1589.7 


201090 


507 


257049 


130323843 


22.5167 


7.9739 


2.70501 


1.97239 


1592.8 


201886 


508 


258064 


131096512 


22.5389 


7.9791 


2.70586 


1.96850 


1595.9 


202683 


509 


259081 


131872229 


22.5610 


7.9843 


2.70672 


1.96464 


1599.1 


202482 


510 


260100 


132651000 


22.5832 


7.9896 


2.70757 


1.96078 


1602.2 


204282 


511 


261121 


133432831 


22.6053 


7.9948 


2.70842 


1.95695 


1605.4 


205084 


512 


262144 


134217728 


22.6274 


8.0000 


2.70927 


1.95312 


1608.5 


205887 


513 


263169 


135005697 


22.6495 


8.0052 


2.71012 


1.94932 


1611.6 


206692 


514 


264196 


135796744 


22.6716 


8.0104 


2.71096 


1.94553 


1614.8 


207499 


515 


265225 


136590875 


22.6936 


8.0156 


2.71181 


1.94175 


1617.9 


208307 


516 


266256 


137388096 


22.7156 


8.0208 


2.71265 


1,93798 


1621.1 


209117 


517 


267289 


138188413 


22.7376 


8.0260 


2.71349 


1.93424 


1624.2 


209928 


518 


268324 


138991832 


22.7596 


8.0311 


2.71433 


1.93050 


1627.3 


210741 


519 


269361 


139798359 


22.7816 


8.0363 


2.71517 


1.92678 


1630.5 


211556 


520 


270400 


140608000 


22.8035 


8.0415 


2.71600 


1.92308 


1633.6 


212372 


521 


271441 


141420761 


22.8254 


8.0466 


2.71684 


1.91939 


1636.8 


213189 


522 


272484 


142236648 


22.8473 


8.0517 


2.71767 


1.91571 


1639.9 


214008 


523 


273529 


143055667 


22.8692 


8.0569 


2.71850 


1.91205 


1643.1 


214829 


524 


274576 


143877824 


22.8910 


8.0620 


2.71933 


1.90840 


1646.2 


215651 


525 


275625 


144703125 


22.9129 


8.0671 


2.72016 


1.90476 


1649.3 


216475 


526 


276676 


145531576 


22.9347 


8.0723 


2.72099 


1.90114 


1652.5 


217301 


527 


277729 


146363183 


22.9565 


8.0774 


2.72181 


1.89753 


1655.6 


218128 


528 


278784 


147197952 


22.9783 


8.0825 


2.72263 


1.89394 


1658.8 


218956 


529 


279841 


148035889 


23.0000 


8.0876 


2.72346 


1.89036 


1661.9 


219787 


530 


280900 


148877000 


23.0217 


8.0927 


2.72428 


1.88679 


1665.0 


220618 


531 


281961 


149721291 


23.0434 


8.0978 


2.72509 


1.88324 


1668.2 


221452 


532 


'.'S3024 


150568768 


23.0651 


8.1028 


2.72591 


1.87970 


1671.3 


222287 


533 


284089 


151419437 


23.0868 


8.1079 


2.72673 


1.87617 


1674.5 


223123 


534 


285156 


152273304 


23.1084 


8.1130 


2.72754 


1.87266 


1677.6 


'223961 


535 


286225 


153130375 


23.1301 


8.1180 


2.72835 


1.86916 


1680.8 


224801 


536 


287296 


153990656 


23.1517 


8.1231 


2.72916 


1.86567 


1683.9 


225642 


537 


288369 


154854153 


23.1733 


8.1281 


2.72997 


1.86220 


1687.0 


226484 


538 


289444 


155720872 


23.1948 


8.1332 


2.73078 


1.85874 


1690.2 


227329 


539 


290521 


156590819 


23.2164 


8.1382 


2.73159 


1.85529 


1693.3 


228175 


540 


291600 


157464000 


23.2379 


8.1433 


2.73239 


1.85185 


1696.5 


229022 


541 


292681 


158340421 


23.2594 


8.1483 


2.73320 


1- 84843 


1699.6 


229871 


542 


293764 


159220088 


23.2809 


8.1533 


2.73400 


1.84502 


1702.7 


230722 


543 


294849 


160103007 


23.3024 


8.1583 


2.73480 


1 84162 


1705.9 


231574 


544 


295936 


160989184 


23.3238 


8.1633 


2.73560 


1.83824 


1709.0 


232428 


545 


297025 


161878625 


23.3452 


8.1683 


2.73640 


1.83486 


1712.2 


233283 


546 


298116 


162771336 


23.3666 


8.1733 


2.73719 


1.88150 


1715.3 


234140 


547 


299209 


163667323 


23.3880 


8.1783 


2.73799 


1.82815 


1718.5 


234998 


548 


300304 


164566592 


23.4094 


8.1833 


2.73878 


1.82482 


1721.6 


235858 


549 


301401 


165469149 


23.4307 


8.1882 


2.73957 


1.82149 


1724.7 


236720 



131 



SQUARES, CUBES, SQUARE ROOTS, CUBE ROOTS, LOGARITHMS, RECIPROCALS, 
CIRCUMFERENCES AND CIRCULAR AREAS OF NOS. FROM 1 T0 1000 



No. 


Square 


Cube 


Sq. Root 


Cu. Root 


Log. 


lOOOxRecip. 


No. = Dia. 




















Circum. 


Area 


550 


302500 


166375000 


23.4521 


8.1932 


2.74036 


1.81818 


1727.9 


237583 


551 


303601 


167284151 


23.4734 


8.1982 


2.74115 


1.81488 


1731.0 


238448 


552 


304704 


168196608 


23.4947 


8.2031 


2.74194 


1.81159 


1734.2 


239314 


553 


305809 


169112377 


23.5160 


8.2081 


2.74273 


1.80832 


1737.3 


240182 


554 


306916 


170031464 


23.5372 


8.2130 


2.74351 


1.80505 


1740.4 


241051 


555 


308025 


170953875 


23.5584 


8.2180 


2.74429 


1.80180 


1743.6 


241922 


556 


309136 


171879616 


23.5797 


8.2229 


2.74507 


1.79856 


1746.7 


242795 


557 


310249 


172808693 


23.6008 


8.2278 


2.74580 


1.79533 


1749.9 


243669 


558 


311364 


173741112 


23.6220 


8.2327 


2.74663 


1.79211 


1753.0 


244545 


559 


312481 


174676879 


23.6432 


8.2377 


2.74741 


1.78891 


1756.2 


245422 


560 


313600 


175616000 


23.6643 


8.2426 


2.74819 


1.78571 


1759.3 


246301 


561 


314721 


176558481 


23.6854 


8.2475 


2.74896 


1.78253 


1762.4 


247181 


552 


315844 


177504328 


23.7065 


8.2524 


2.74974 


1.77936 


1765.6 


248063 


563 


316969 


178453547 


23.7276 


8.2573 


2.75051 


1.77620 


1768.7 


248947 


564 


318096 


179406144 


23.7487 


8.2621 


2.75128 


1.77305 


1771.9 


249832 


565 


319225 


180362125 


23.7697 


8.2670 


2.75205 


1.76991 


1775.0 


250719 


566 


320a56 


181321496 


23.7908 


8.2719" 


2.75282 


1.76678 


1778.1 


251607 


567 


321489 


182284263 


23.8118 


8.2768 


2.75358 


1.76367 


1781.3 


252497 


568 


322621 


183250432 


23.8328 


8.2816 


2.75435 


1.76056 


1784.4 


253388 


569 


323761 


184220009 


23.8537 


8.2865 


2.75511 


1.75747 


1787.6 


254281 


570 


324900 


185193000 


23.8747 


8.2913 


2.75587 


1.75439 


1790.7 


255176 


571 


326041 


186169411 


23.8956 


8.2962 


2.75664 


1.75131 


1793.9 


256072 


572 


327184 


187149248 


23.9165 


8.3010 


2.75740 


1.74825 


1797.0 


256970 


573 


328329 


188132517 


23.9374 


8.3059 


2.75815 


1.74520 


1800.1 


257869 


574 


329476 


189119224 


23.9583 


8.3107 


2.75891 


1.74216 


1803.3 


258770 


575 


330625 


190109375 


23.9792 


8.3155 


2.75967 


1.73913 


1806.4 


259672 


576 


331776 


191102976 


24.0000 


8.3203 


2.70042 


1.73611 


1809.6 


260576 


577 


332929 


192100033 


24.0208 


8.3251 


2.76118 


1.73310 


1812.7 


261482 


578 


334084 


193100552 


24.0416 


8.3300 


2.76193 


1.73010 


1815.8 


- 262389 


579 


335241 


194104539 


24.0624 


8.3348 


2.76268 


1.72712 


1819.0 


263298 


580 


336400 


195112000 


24.0832 


8.3396 


2.76343 


1.72414 


1822.1 


264208 


581 


337561 


196122941 


24.1039 


8.3443 


2.76418 


1.72117 


1825.3 


265120 


582 


338724 


197137368 


24.1247 


8.3491 


2.76492 


1.71821 


1828.4 


2660:33 


583 


339889 


198155287 


24.1454 


8.3539 


2.76567 


1.71527 


1831.6 


266948 


584 


341056 


199176704 


24.1661 


8.3587 


2.76641 


1.71233 


1834.7 


267865 


585 


342225 


200201625 


24.1868 


8.3634 


2.76716 


1.70940 


1837.8 


268783 


586 


343396 


201230056 


24.2074 


8. 3682 


2.76790 


1.70649 


1841.0 


269701 


587 


344569 


202262003 


24.2281 


8.3730 


2.76864 


1.70358 


1844.1 


270624 


588 


345744 


203297472 


24.2487 


8.3777 


2.76938 


1.70068 


1847.3 


271547 


689 


346921 


204336469 


24.2693 


8.3825 


2.77012 


1.69779 


1850.4 


272471 


590 


348100 


205379000 


24.2899 


8.3872 


2.77085 


1.69492 


1853.5 


273397 


591 


349281 


206425071 


24.3105 


8.3919 


2.77159 


1.69205 


1856.7 


274325 


592 


350464 


207474688 


24.3311 


8.3967 


2 77232 


1.68919 


1859.8 


275254 


593 


351649 


208527857 


24.3516 


8.4014 


2.77305 


1.6S634 


1863.0 


276184 


594 


352836 


209584584 


24.3721 


8.4061 


2.77379 


1.68350 


1866.1 


277117 


595 


354025 


210644875 


24.3926 


8.4108 


2.77452 


1.6S067 


1869.3 


278051 


596 


355216 


211708736 


24.4131 


8.4155 


2.77525 


1.67785 


1872.4 


278986 


597 


856409 


212776173 


24.4336 


8-4202 


2.77597 


1.67504 


1875.5 


279923 


598 


357604 


213847192 


24.4540 


8.4249 


2.77670 


1.67224 


1878.7 


280862 


599 


358S01 


214921799 


24.4745 


8.4296 


2.77743 


1.66945 


1881.8 


281802 



132 



SQUARES, CUBES, SQUARE ROOTS, CUBE ROOTS, LOGARITHMS, RECIPROCALS, 
CIRCUMFERENCES AND CIRCULAR AREAS OF NOS. FROM 1 TO 1000 



tfn 




Cube 


Sq. Root 


Cu. Root 


Log. 


lOOOxRecip. 


No. = Dia. 


iiO. u^uaiu 


















Circum. 


Area 


600 360000 


216000000 


24.4949 


8.4343 


2.77815 


1.66667 


1885.0 


282743 


601 


361201 


217081801 


24.5153 


8.4390 


2.77887 


.1.66389 


1888.1 


283687 


602 


362404 


218167208 


24.5357 


8.4437 


2.77960 


1.66113 


1891.2 


284631 


603 


3636H9 


219256227 


24.5561 


8.4484 


2.78032 


1.65837 


1894.4 


285578 


604 


364*10 


220348864 


24.5764 


8.4530 


2.78104 


1.65563 


1897.5 


286526 


605 


3660-25 


221445125 


24.5967 


8.4577 


2.78176 


1.65289 


1900.7 


287475 


606 


367236 


222545010 


24.6171 


8.4623 


2.78247 


1.65017 


1903.8 


288426 


607 . 368449 


2236485-13 


24.6374 


8.4670 


2.78319 


1.64745 


1907.0 


289379 


608 | 369664 


224755712 


24.6577 


8.4716 


2.78390 


1.64474 


1910.1 


290333 


609 370881 


225866529 


24.6779 


8.4763 


2.78462 


1.64204 


1913.2 


291289 


610 ! 372100 


226981000 


24.6982 


8.4809 


2.78533 


1.63934 


1916.4 


292247 


611 | 373321 


228099131 


24.7184 


8.4850 


2.78604 


1.63666 


1919.5 


293206 


612 374544 


229220928 


24.7386 


8.4902 


2.78675 


1.63399 


1922.7 


294166 


613 375769 


2303461397 


24.7588 


8.4948 


2.78746 


1.63132 


1925.8 


295128 


614 ! 376996 


231475544 


24.7790 


8.4994 


2.78817 


1.62866 


1928.9 


296092 


615 378225 


232608375 


24.7992 


8.5040 


2.78888 


1.62602 


1932.1 


297(157 


616 379456 


233744896 


24.8193 


8.5086 


2.78958 


1.62338 


1935.2 


298024 


617 380689 


234885113 


24.8395 


8.5132 


2.79029 


1.62075 


1938.4 


298992 


618 381924 


236,u29(i32 


24.8596 


8.5178 


2.79099 


1.61812 


1941-5 


299962 


619 ; 383161 


237170059 


24.8797 


8.5224 


2.79169 


1.61551 


1944.7 


300934 


620 


384400 


238328000 


24.8998 


8.5270 


2.79239 


1.61290 


1947.8 


301907 


621 


385641 


2394.83061 


24.9199 


8.5316 


2.70309 


1.61031 


1950.9 


302**2 


622 


38688-4 


240641848 


24.9399 


8.5302 


2.79379 


1.60772 


1954.1 


303*5* 


623 


3S8129 


241804367 


24.9600 


8.5408 


2.79449 


1.60514 


1957.2 


304836 


624 389376 


242970024 


21.9800 


8.5453 


2.79518 


1.60256 


1960.4 


305815 


625 390625 


244140625 


25.0000 


8.5499 


2.79588 


1.60000 


1963.5 


306796 


626 1 391876 


245314370 


25.0200 


8.5544 


2.79657 


1.59744 


1966.6 


3077T9 


627 ; 393129 


246491883 


25.04IX) 


.8.5590 


2.79727 


1.59490 


1969.8 


308763 


628 394384 


247073152 


25.0599 


8.5635 


2.79796 


1.59236 


1972.9 


309748 


629 395641 


248858189 


25.0799 


8.5681 


2.79865 


1.58983 


1976.1 


310736 


630 


396900 


250047000 


25.0998 


8.5726 


2.79934 


1.58730 


1979.2 


311725 


631 


398101 


251239591 


25.1197 


8.5772 


2.80003 


1.58479 


1982.4 312715 


632 


399424 


2524^35908 


25.1396 


8.5817 


2.80072 


1.58228 


1985.5 


313707 


633 


400689 


253036137 


25.1595 


8.5862 


2.80140 


1.57978 


1988.6 


314700 


•534 


401950 


254840104 


25.1794 


8.5907 


2.80209 


1.57729 


1991.8 


315696 


635 


403225 


256047875 


25.1992 


8.5952 


2.80277 


1.57480 


1994.9 


316692 


636 


404490 


257259456 


25.2190 


8.5997 


2.80346 


1.57233 


1998.1 


317690 


'•.37 


405769 


258474853 


25.2389 


8.6043 


2.80414 


1.56986 


2001.2 


318690 


638 


407044 


259694072 


25.25*7 


8.6088 


2.80482 


1.56740 


2004,3 


319692 


639 


408321 


260917119 


25.2784 


8.6132 


2.80550 


1.56495 


2007.5 320695 


640 


409600 


202144000 


25.2982 


8.6177 


2.80618 


1.56250 


2010.6 ! 321699 


641 


410881 


203374721 


25.3180 


8.6222 


2.80686 


1.56006 


2013.8 322705 


642 


412164 


20460928S 


25.3377 


8.6267 


2.80754 


1.55763 


2016.9 


323713 


643 


413449 


265847707 


25.3574 


8.6312 


2.80821 


1.55521 


2020.0 


324722 


644 


414736 


267089984 


25.3772 


8.6357 


2.80889 


1.55280 


2023.2 


325733 


645 


416025 


268336125 


25.3969 


8.6401 


2.80956 


1.55039 


2026.3 


326745 


646 


417316 


209586130 


25.4165 


8.6446 


2.81023 


1.54799 


2029.5 


327759 


017 


418609 


270840023 


25.4362 


8.6490 


2.81090 


1.54560 


2032.6 


328775 


648 


419904 


272097792 


25 4558 


8.6535 


2.81158 


1.54321 


2035.8 


329792 


649 


421201 


273:359449 


25.4755 


8.6579 


2.81224 


1.54083 


2038.9 


330810 



133 



SQUARES, CUBES, SQUARE ROOTS, CUBE ROOTS, LOGARITHMS, RECIPROCALS, 
CIRCUMFERENCES AND CIRCULAR AREAS OF NOS. FROM 1 TO 1000 



> T o. 


Square 


Cube 


Sq. Root 


Gu. Root 


Log. 


lOOOxRecip. 


Ho. = Dia. 


















Circum. 


Area 


050 


422500 


274025000 


25.4951 


8.6624 


2.81291. 


1.53846 


2042.0 


331831 


651 


423S01 


275894451 


25.5147 


8.6668 


2.81358 


1.53610 


2045.2 


332853 


(152 


425104 


277167808 


25.5343 


8.6713 


2.81425 


1.53374 


2048.3 


333S76 


653 


426409 


278445077 


25.5539 


8.6757 


2.81491 


1.53139 


2051.5 


334901 


'154 


427716 


279726264 


25.57:34 


8.6801 


2.81558 


1.52905 


2054.6 


335927 


055 


429025 


281011375 


25.5930 


8.0,845 


2.81624 


1.52672 


2057.7 


336955 


(356 


430336 


282300416 


25.6125 


8.0890 


2.81690 


1.52439 


2060.9 


337985 


057 


431649 


283593393 


25.6320 


8.6934 


2.81757 


1.52207 


2064.0 


339016 


058 


432964 


284890312 


25.6515 


8.0978 


2.81823 


1.51976 


2067.2 


340049 


059 


434281 


286191179 


25.6710 


8.7022 


2.81889 


1.51745 


2070.3 


341084 


660 


435600 


287496000 


25.6905 


8.7066 


2.81954 


1.51515 


2073.5 


342119 


661 


436921 


288804781 


25.7099 


8.7110 


2.82020 


1.51286 


2076.6 


343157 


002 


438244 


290117528 


25.7294 


8.7154 


2.82086 


1.51057 


2079.7 


344196 


003 


439569 


291434247 


25.7488 


8.7198 


2.82151 


1.50830 


2082.9 


345237 


004 


440896 


292754944 


25.7083 


8.7241 


2.82217 


1.50603 


2086.0 


346279 


005 


442225 


294079625 


25.7876 


8.7285 


2.82282 


1.50376 


20S9.2 


347323 


606 


443556 


295408296 


25.6070 


8.7329 


2.82347 


1.501 50 


2092.3 


348368 


667 


444889 


296740963 


25.8263 


8.7373 


2.82413 


1.49925 


2095.4 


349415 


668 


446224 


298077632 


25.8457 


8.7416 


2.82478 


1.49701 


2098.6 


350464 


669 


447561 


299418309 


25.8650 


8.7460 


2.82543 


1.49477 


2101.7 


351514 


670 


448900 


300763000 


25.8844 


8.7503 


2.82607 


1.49254 


2104.9 


352565 


671 


450241 


302111711 


25.9037 


8.7547 


2.82072 


1 .49031 


2108.0 


353018 


672 


451 584 


303464448 


25.9230 


8.7590 


2.S2737 


1.48810 


2111.2 


354673 


673 


452929 


304821217 


25.9422 


8.76:34 


2.82802 


1.48588 


2114.3 


355730 


674 


454276 


306182024 


25.9615 


8.7677 


2.82866 


1.48368 


2117.4 


356788 


675 


455625 


307546875 


25.9808 


8.7721 


2.82930 


1.48148 


2120.6 


357847 


676 


456976 


308915776 


26.0000 


8.7764 


2.82995 


1.47929 


2123.7 


358908 


677 


458329 


310288733 


26.0192 


8.7807 


2.83059 


1.47711 


2126.9 


359971 


678 


459684 


311665752 


26.0:384 


8.7850 


2.83123 


1.47493 


2130.0 


'301035 


679 


461041 


313046839 


26.0576 


8.7893 


2.83187 


1.47275 


2133.1 


362101 


680 


462400 


314432000 


26.0768 


8.7937 


2.83251 


1.47059 


2136.3 


363168 


681 


463761 


315821241 


26.0960 


8.7P80 


2.83315 


1.46843 


2139.4 


304237 


6S2 


465124 


317214568 


26.1151 


8.8023 


2.83378 


1.46628 


2142.6 


305308 


683 


466489 


318611987 


26.1343 


8.8006 


2.83442 


1.46413 


2145.7 


300380 


684 


467856 


320013504 


26.1534 


8.8109 


2.83506 


1.46199 


2148.9 


807453 


685 


469225 


321419125 


26.1725 


8.8152 


2.83569 


1.45985 


2152.0 


3I1K52* 


686 


470596 


322828856 


26.1916 


8.8194 


2.83632 


1.45773 


2155.1 


309005 


687 


471969 


324242703 


26.2107 


8.8237 


2.83696 


1.45560 


2158.3 


370684 


688 


473344 


325660672 


26.2298 


8.8280 


2.83759 


1.45349 


2101.4 


371704 


689 


474721 


3270S2769 


26.2488 


8.8323 


2.83822 


1.45138 


2164.6 


372845 


690 


476100 


328509000 


26.2679 


8.8366 


2.83885 


1.44928 


2167.7 


373928 


691 


477481 


329930371 


26.2869 


8.8408 


2.83948 


1.44718 


2170.8 


375013 


692 


478864 


331373888 


26.3059 


8.8451 


2.84011 


1.44509 


2174.0 


376099 


693 


480249 


332812557 


26.3249 


8.8493 


2.84073 


1.44300 


2177.1 


377187 


694 


481636 


334255384 


26.3439 


8.8536 


2.84136 


1.44092 


2180.3 


378276 


695 


483025 


335702375 


26.3629 


8.8578 


2.84198 


1.43885 


2183.4 


379367 


696 


484416 


337153536 


26.3818 


8.8621 


2.84261 


1.43678 


2186.6 


380459 


697 


485809 


338608873 


20.4008 


8.8663 


2.84323 


1.43472 


2189.7 


881554 


698 


487204 


340068392 


26.4197 


8.8706 


2.84386 


1.43267 


2192.8 


382049 


699 


488601 


341532099 


26.4386 


8.8748 


2.84448 


1.43062 


2196.0 


883746 



134 



SQUARES, CUBES, SQUARE ROOTS, CUBE ROOTS, LOGARITHMS, RECIPROCALS, 
CIRCUMFERENCES AND CIRCULAR AREAS OF NOS. FROM 1 TO 1000 



No. 


Square 


Cube 


Sq. Root 


Cu. Root 


Log. 


lOOOiRecip. 


Ho.= 


Dia. 


Circum. 


Area 


700 


490000 


343000000 


26.4575 


8.8790 


2.84510 


1.42857 


2199.1 


384845 


701 


491401 


344472101 


26.4764 


8.8833 


2.84572 


1.42053 


2202.3 


385945 


702 


492804 


345948408 


26.4953 


8.8875 


2.84634 


1.42450 


2205.4 


387047 


703 


494209 


347428927 


26.5141 


8.8917 


2.84696 


1.42248 


2208.5 


388151 


704 


495616 


348913664 


26.5330 


8.8959 


2.84757 


1.42046 


2211.7 


389256 


705 


497025 


350402625 


26.5518 


8.9001 


2.84819 


1.41844 


2214.8 


390303 


706 


498436 


351895816 


26.5707 


8.9043 


2.84880 


1.41643 


2218.0 


391471 


107 


499849 


353:393243 


26.5895 


8.9085 


2.84942 


1.41443 


2221.1 


392580 


708 


501264 


354894912 


26.0083 


8.9127 


2.85003 


1.41243 


2224.3 


393692 


709 


502681 


356400829 


26.6271 


8.9169 


2.85065 


1.41044 


2227.4 


394805 


710 


504100 


357911000 


26.6458 


8.9211 


2.85126 


1.40845 


2230.5 


395919 


711 


505521 


359425431 


26.6046 


8.9253 


2.85187 


1.40047 


22:33.7 


397035 


712 


506944 


360944128 


26.6833 


8.9295 


2.85248 


1.40449 


2236.8 


398153 


713 


508369 


362407097 


26.7021 


8.9337 


2.85309 


1.40253 


2240.0 


399272 


714 


509796 


363994344 


26.7208 


8.9378 


2.85370 


1.40056 


2243.1 


400393 


715 


511225 


305525875 


20.7395 


8.9420 


2.85431 


1.39860 


2246.2 


401515 


716 


512656 


367061696 


26.7582 


8.9402 


2.85491 


1.39665 


2249.4 


402639 


717 


514089 


368601813 


26.7709 


8.9503 


2.85552 


1.39470 


2252.5 


403765 


718 


515524 


370146232 


26.7955 


8.9545 


2.85612 


1.39276 


2255.7 


404892 


719 


516961 


371694959 


26.8142 


8.9587 


2.85673 


1.39082 


2258.8 


406020 


720 


518400 


373248000 


26.8328 


8.9628 


2.85733 


1.38889 


2261.9 


40*150 


721 


519841 


374805301 


26.8514 


8.9670 


2.85794 


1.38096 


2265.1 


4U8282 


722 


521284 


376367048 


26.8701 


8.9711 


2.85854 


1.38504 


2268.2 


409416 


723 


522729 


3779:33067 


26.8887 


8.9752 


2.85914 


1.38313 


2271.4 


410550 


724 


524176 


379503424 


26.9072 


8.9794 


2.85974 


1.38122 


2274.5 


411687 


725 


525625 


381078125 


20.9258 


8.9835 


2.86034 


1.37931 


2277.7 


412825 


726 


527076 


382657176 


26.9444 


8.9876 


2.86094 


1.37741 


2280.8 


413965 


727 


528529 


384240583 


26.9629 


8.9918 


2.80153 


1.37552 


2283.9 


415106 


128 


529984 


385828352 


26.9815 


8.9959 


2.80213 


1.37363 


2287.1 


416248 


729 531441 


387420489 


27.0000 


9.0000 


2.86273 


1.37174 


2290.2 


417393 


730 


532900 


389017000 


27.0185 


9.0041 


2.86332 


1.36986 


2293.4 


418539 


731 


534361 


390617891 


27.0370 


9.0082 


2.80392 


1.36799 


2296.5 


419686 


732 


535824 


392223168 


27.0555 


9.0123 


2.86451 


1.36612 


2299.7 


420835 


733 


537289 


393832837 


27.0740 


9.0104 


2.86510 


1.36426 


2302.8 


421986 


734 


538756 


895446904 


27.0924 


9.0205 


2.86570 


1.36240 


2305.9 


423138 


735 


540225 


397065375 


27.1109 


9.0246 


2.86629 


1.30054 


2309.1 


424293 


736 


541096 


398088250 


27.1293 


9.0287 


2.80688 


1.35870 


2312.2 


425448 


7 3 7 


543169 


400315553 


27.1477 


9.0.328 


2.86747 


1.35685 


2315.4 


426604 


738 


544644 


401947272 


27.1662 


9.0309 


2.80800 


1.35501 


2.318.5 


427762 


739 


546121 


403583419 


27.1846 


9.0410 


2.86864 


1.35318 


2321.6 


428922 


710 


517600 


405224000 


27.2029 


9.0450 


2.86923 


1.35135 


2324.8 


430084 


741 


549081 


406869021 


27.2213 


9.0491 


2.80982 


1.34953 


2327.9 


431247 


742 


5505(34 


408518488 


27.2307 


9.0532 


2.87040 


1.34771 


2331.1 


432412 


743 


552049 


410172407 


27.2580 


9.0572 


2.87099 


1.34590 


2334.2 


43:3578 


744 


553536 


411830784 


27.2764 


9.0013 


2.87157 


1.34409 


2337.3 


434746 


745 


555025 


413493025 


27.2947 


9.0054 


2.87216 


1.84228 


2340.5 


435916 


746 


556516 


415100936 


27.3130 


9.0094 


2.87274 


1.34048 


2343.6 


437087 


747 
748 


658009 


416832723 


27.3313 


9.0735 


2.87332 


1.33869 


2846.8 


438259 


559504 


418508992 


27.3496 


9.0775 


2.87390 


1.33690 


2349.9 


439438 


749 


561001 


420189749 


27.3079 


9.0816 


2.87448 


1.33511 


2353.1 


440609 



135 



SQUARES, CUBES, SQUARE ROOTS, CUBE ROOTS, LOGARITHMS, RECIPROCALS, 
CIRCUMFERENCES AND CIRCULAR AREAS OF NOS. FROM 1 TO 1000 



No. 


Square 


Cube 


Sq.Hoot 


Cu. Root 


Log. 


lOOOxRecip. 


No. = 


Dia, 




Circum. 


Area 


750 


562500 


421875000 


27.3861 


9.0856 


2.87506 


1.33333 


2356.2 


441786 


751 


564001 


423564751 


27.4044 


9.0896 


2.87564 


1.33156 


2359.3 


442965 


752 


565504 


425259008 


27.4226 


9.0937 


2.87622 


1.32979 


2362.5 


444146 


753 


567009 


426957777 


27.4408 


9.0977 


2.87680 


1.32802 


2365.6 


445328 


754 


568516 


428661064 


27.4591 


9.1017 


2.87737 


1.32626 


2368.8 


446511 


755 


570025 


430368875 


27.4773 


9.1057 


2.87795 


1.32450 


2371.9 


447697 


756 


571536 


432081216 


27.4955 


9.1098 


2.87852 


1.32275 


2375.0 


448883 


757 


573049 


433798093 


27.5136 


9.1138 


2.87910 


1.32100 


2378.2 


450072 


758 


574564 


435519512 


27.5318 


9.1178 


2.87967 


1.31926 


2381.3 


451262 


759 


576081 


437245479 


27.5500 


9.1218 


2.88024 


1.31752 


2384.5 


452453 


760 


577600 


438976000 


27.5681 


9.1258 


2.88081 


1.31579 


2387.6 


453646 


761 


579121 


440711081 


27.5862 


9.1298 


2.88138 


1.31406 


2390.8 


454841 


762 


580644 


442450728 


27.6043 


9.1338 


2.88196 


1.31234 


2393.9 


456037 


763 


582169 


444194947 


27.6225 


9.1378 


2.88252 


1.31062 


2397.0 


457234 


764 


583696 


445943744 


27.6405 


9.1418 


2.88309 


1.30890 


2400.2 


458434 


765 


585225 


447697125 


27.6586 


9.1458 


2.88366 


1.30719 


2403.3 


459635 


766 


586756 


449455096 


27.6767 


9.1498 


2.88423 


1.30548 


2406.5 


460837 


767 


588289 


451217663 


27.6948 


9.1537 


2.88480 


1.30378 


2409.6 


462042 


768 


589824 


452984832 


27.7128 


9.1577 


2.88536 


1.30208 


2412.7 


463247 


769 


591361 


454756609 


27.7308 


9.1617 


2.88593 


1.30039 


2415.9 


464454 


770 


593900 


456533000 


27.7489 


9.1657 


2.88649 


1.29870 


2419.0 


465663 


771 


594441 


458314011 


27.7669 


9.1696 


2.88705 


1.29702 


2422.2 


466873 


772 


595984 


460099648 


27.7849 


9.1736 


2.88762 


1.29534 


2425.3 


468085 


773 


597529 


461889917 


27.8029 


9.1775 


2/. 88818 


1.29366- 


2428.5 


469298 


774 


599076 


463684824 


27.8209 


9.1815 


2.88874 


1.29199 


2431.6 


470513 


775' 


600625 


465484375 


27.8388 


9.1855 


2.88930 


1.29032 


2434.7 


471730 


776 


602176 


467288576 


27.8568 


9.1894 


2.88986 


1.28866 


2437.9 


472948 


777 


603729 


469097433 


27.8747 


9.1933 


2.89042 


1.28700 


2441.0 


474168 


778 


605284 


470910952 


27.8927 


9.1973 


2.89098 


1.28535 


2444.2 


475389 


779 


606841 


472729139 


27.9106 


9.2012 


2.89154 


1.28370 


2447.3 


476612 


780 


608400 


474552000 


27.9285 


9.2052 


2.89209 


1.28205 


24.50.4 


477836 


781 


609961 


476379541 


27.9464 


9.2091 


2.89265 


1.28041 


2453.6 


479062 


782 


611524 


478211768 


27.9643 


9.2130 


2.89321 


1.27877 


2456.7 


480290 


783 


613089 


480048687 


27.9821 


9.2170 


2.89376 


1.27714 


2459.9 


481519 


784 


614656 


481890304 


28.0000 


9.2209 


2.89432 


1.27551 


2463.0 


482750 


785 


616225 


483736625 


28.0179 


9.2248 


2.89487 


1.27389 


2466.2 


483982 


,786 


617796 


485587656 


28.0357 


9.2287 


2.89542 


1.27226 


2469.3 


485216 


787 


619369 


487443403 


28.0535 


9.2326 


"2.89597 


1.27065 


2472.4 


486451 


788 


620944 


489303872 


28.0713 


9.2365 


2.89653 


1.26904 


2475.6 


487688 


789 


622521 


491169069 


28.0891 


9.2404 


2.89708 


1.26743 


2478.7 


488927 


790 


624100 


493039000 


28.1069 


9.2443 


2.89763 


1.26582 


2481.9 


490167 


791 


625681 


494913671 


28.1247 


9.2482 


2.89818 


1.26422 


2485.0 


491409 


792 


627264 


4967930S8 


28.1425 


9.2521 


2.89873 


1.26263 


2488.1 


492652 


793 


628849 


498677257 


28.1603 


9.2560 


2.89927 


1.26103 


2491.3 


493897 


794 


630436 


500566184 


28.1780 


9.2599 


2.89982 


1.25945 


2494.4 


495143 


795 


632025 


502459875 


28.1957 


9.2638 


2.90037 


1.25786 


2497.6 


496391 


796 


633616 


504358336 


28.2135 


9.2677 


2.90091 


1.25628 


2500.7 


497641 


797 


635209 


506261573 


28.2312 


9.2716 


2.90146 


1.25471 


2503.8 


498892 


798 


636804 


508169592 


28.2489 


9.2754 


2.90200 


1.25313 


2507.0 


500145 


799 


638401 


510082399 


28.2666 


9.2793 


2.90255 


1.25156 


2510.1 


501399 



136 



SQUARES, CUBES, SQUARE ROOTS, CUBE ROOTS, LOGARITHMS, RECIPROCALS, 
CIRCUMFERENCES AND CIRCULAR AREAS OF NOS. FROM 1 TO 1000 

















No. = Dia. 


No. 


Square 


Cube 


Sq. Root 


Cu. Root 


Log. 


iOOOxRecip. 




Circum. I Area 


800 


640000 


512000000 


28.2843 


9.2832 


2.90309 


1.25000 


2513.3 


502655 


801 


641601 


513922401 


28.3019 


9.2870 


2.90363 


1.24844 


2516.4 


503912 


802 


643204 


515849608 


28.3196 


9.2909 


2.90417 


1.24688 


2519.6 


505171 


803 


644809 


517781627 


28.3373 


9.2948 


2.90472 


1.24533 


2522.7 


506432 


804 


646416 


519718464 


28.3549 


9.2986 


2.90526 


1.24378 


2525.8 


507694 


805 


648025 


521660125 


28.3725 


9.3025 


2.90580 


1.24224 


2529.0 


508958 


806 


649636 


523606616 


28.3901 


9.3063 


2.90634 


1.24069 


2532.1 


510223 


80? 


651249 


525557943 


28.4077 


9.3102 


2.9068? 


1.23916 


2535.3 


511490 


808 


652864 


527514112 


28.4253 


9.3140 


2.90741 


1.23762 


2538.4 


512758 


809 


654481 


529475129 


28.4429 


9.3179 


2.90795 


1.23609 


2541.5 


514028 


810 


656100 


531441000 


28.4605 


9.3217 


2.90849 


1.23457 


2544.7 


515300 


811 


657721 


53:3411731 


28.4781 


9.3255 • 


2.90902 


1.23305 


2547.8 


516573 


812 


659344 


535387328 


28.4956 


9.3294 


2.90956 


1.23153 


2551.0 


517848 


-813 


660969 


537367797 


28.5132 


9.3332 


2.91009 


1.23001 


2554.1 


519124 


814 


662596 


539353144 


28.5307 


9.3370 


2.91062 


1.22850 


2557.3 


520402 


815 


664225 


541343375 


28.5482 


9.3408 


2.91116 


1.22699 


2560.4 


521681 


816 


665856 


543338496 


28.5657 


9.3447 


2.91169 


1.22549 


2563.5 


522962 


817 


667489 


545338513 


28.5832 


9.3485 


2.91222 


1.22399 


2566.7 


524245 


818 


669124 


547343432 


28.6007 


9.3523 


2.91275 


1.22249 


2569.8 


525529 


819 


670761 


549353259 


28.6182 


9.3561 


2.91328 


1.22100 


2573.0 


526814 


820 


672400 


551368000 


28.6356 


9.3599 


2.91381 


1.21951 


2576.1 


528102 


821 


674041 


553387661 


28.6531 


9.3637 


2.91434 


1.21803 


2579.2 


529391 


822 


675684 


555412248 


28.6705 


9.3675 


2.91487 


1.21655 


2582.4 


530681 


823 


677329 


557441767 


28.6880 


9.8713 


2.91540 


1.21507 


2585.5 


531973 


824 


678976 


559476224 


28.7054 


9.3751 


2.91593 


1.21359 


2588.7 


533267 


825 


6S0625 


561515625 


28.7228 


9.3789 


2.91645 


1.21212 


2591.8 


534562 


826 


682276 


563559976 


28.7402 


9.3827 


2.91698 


1.21065 


2595.0 


535858 


827 


683929 


565609283 


28.7576 


9.3865 


2.91751 


1.20919 


2598.1 


537157 


828 


6855,84 


567663552 


28.7750 


9.3902 


2.91803 


1.20773 


2601.2 


538456 


829 687241 


569722789 


28.7924 


9.3940 


2.91855 


1.20627 


2604.4 


539758 


830 | 688900 


571787000 


28.8097 


9.3978 


2.91908 


1.20482 


2607.5 


541061 


831 ! 690561 


573856191 


28.8271 


9.4016 


2.91960 


1.20337 


2610.7 


542365 


832 | 692224 


5759303GS 


28.8444 


9.4053 


2.92012 


1.20192 


2613.8 


543671 


833 ; 693889 


57.8009537 


28.8617 


9.4091 


2.92065 


1.20048 


2616.9 


544979 


834 | 695556 


580093704 


28.8791 


9.4129 


2.92117 


1.19904 


2620.1 


546288 


835 • 697225 


582182875 


28.8964 


9.4166 


2.92169 


1.19760 


2623.2 


547599 


836 1 698896 


584277056 


28.9137 


9.4204 


2.92221 


1.19617 


2626.4 


548912 


837 ; 700569 


5SG376253 


28.9310 


9.4241 


2.92273 


1.19474 


2629.5 


550226 


838 | 702244 


588480472 


28.9482 


9.4279 


2.92324 


1.19332 


2632.7 


551541 


839 703921 


590589719 


28.9655 


9.4316 


2.92376 


1.19189 


2635.8 


552858 


840 ! 705600 


592704000 


28.9828 


9.4354 


2.92428 


1.19048 


2638.9 


554177 


841 l 70?28l 


59482*321 


29.0000 


9.4391 


2.92480 


1.18906 


2642.1 


555497 


842 708964 


596947688 


29.0172 


9.4429 


2.92531 


1.18765 


2645.2 


556819 


843 


710649 


599077107 


29.0345 


9.4466 


2.92583 


1 . 18624 


2648.4 


558142 


844 


712336 


601211584 


29.0517 


9.4503 


2.92634 


1.18483 


2651.5 


559467 


845 


714025 


603351125 


29.0689 


9.4541 


2.92686 


1.18343 


2654.6 


560794 


846 


715716 


605495736 


29.0861 


9.4578 


2.92737 


1.18203 


2657.8 


562122 


847 


717409 


607645423 


29.1033 


9.4615 


2.92788 


1.18064 


2660.9 


563452 


848 


719104 


609800192 


29.1204 


9.4652 


2.92840 


1 . 17925 


2664.1 


564783 


849 


720801 


611960049 


29.1376 


9.4690 


2.92891 


1.17786 


2667.2 


566116 



137 



SQUARES, CUBES, SQUARE ROOTS, CUBE ROOTS, LOGARITHMS, RECIPROCALS, 
CIRCUMFERENCES AND CIRCULAR AREAS OF NOS. FROM 1 T0 1000 



No. 


Square 


Cube 


Sq. Root 


Cu. Root 


Log. 


lOOOxRecip. 


No. = Dia. 






Circum. 


Area 


850 


722500 


614125000 


29.1548 


9.4727 


2.92942 


1.17647 


2670.4 


567450 


851 


724201 


616295051 


29.1719 


9.4764 


2.92993 


1.17509 


2673.5 


568786 


852 


725904 


618470208 


29.1890 


9.4801 


2.93044 


1.17371 


2676.6 


570124 


853 


727609 


620650477 


29.2002 


9.4838 


2.93095 


1.17233 


2679.8 


571463 


854 


729316 


622835864 


29.2233 


9.4875 


2.93146 


1.17096 


2682.9 


572803 


855 


731025 


625026375 


29.2404 


9.4912 


2.93197 


1.16959 


2686.1 


574146 


856 


732736 


627222016 


29.2575 


9.4949 


2.93247 


1.16822 


2689.2 


575490 


857 


734449 


629422793 


29.2746 


9.4986 


2.93298 


1.16686 


2692.3 


576885 


858 


736164 


631628712 


29.2916 


9.5023 


2.93349 


1.16550 


2695.5 


578182 


859 


737881 


633839779 


29.3087 


9.5060 


2.93399 


1.16414 


2698.6 


579530 


860 


739600 


636056000 


29.3258 


9.5097 


2.93450 


1.16279 


2701.8 


580880 


861 


741321 


638277381 


29.3428 


9.5134 


2.93500 


1.16144 


2704.9 


582232 


862 


743044 


640503928 


29.3598 


9.5171 


2.93551 


1.16009 


2708.1 


583585 


863 


744769 


642735647 


29.3769 


9.5207 


2.93601 


1.15875 


2711.2 


584940 


864 


746496 


644972544 


29.3939 


9.5244 


2.93651 


1.15741 


2714.3 


586297 


865 


748225 


647214625 


29.4109 


9.5281 


2.93702 


1.15607 


2717.5 


587655 


866 


749956 


649461896 


29.4279 


9.5317 


2.93752 


1.15473 


2720.6 


589014 


867 


751689 


651714363 


29.4449 


9.5354 


2.93802 


1.15340 


2723.8 


590375 


868 


753424 


653972032 


29.4618 


9.5391 


2.93852 


1.15207 


2726.9 


591738 


869 


755161 


656234909 


29.4788 


9.5427 


2.93902 


1.15075 


2730.0 


593102 


870 


756900 


658503000 


29.4958 


9.5464 


2.93952 


1.14943 


2733.2 


594468 


871 


758641 


660776311 


29.5127 


9.5501 


2.94002 


1.14811 


2736.3 


595835 


872 


760384 


663054848 


29.5296 


9.5537 


2.94052 


1.14679 


2739.5 


597204 


873 


762129 


665338617 


29.5466 


9.5574 


2.94101 


1.14548 


2742.6 


598575 


874 


763876 


667627624 


29.5635 


9.5610 


2.94151 


1.14416 


2745.8 


599947 


875 


765625 


669921875 


29.5804 


9.5647 


2.94201 


1.14286 


2748.9 


601320 


876 


767376 


672221376 


29.5973 


9.5683 


2.94250 


1.14155 


2752.0 


602696 


877 


769129 


674526133 


29.6142 


9.5719 


2.94300 


1.14025 


2755.2 


604073 


878 


770884 


676836152 


29.6311 


9.5756 


2.94349 


1.13895 


2758.3 


605451 


879 


772641 


679151439 


29.6479 


9.5792 


2.94399 


1.13766 


2761.5 


606831 


880 


774400 


681472000 


29.6648 


9.5828 


2.94448 


1.13636 


2764.6 


608212 


881 


776161 


683797841 


29.6816 


9.5865 


2.94498 


1.13507 


2767.7 


609595 


882 


777924 


686128968 


29.6985 


9.5901 


2.94547 


1.13379 


2770.9 


610980 


883 


779689 


688465387 


29.7153 


9.5937 


2.94596 


1.13250 


2774.0 


612366 


884 


781456 


690807104 


29.7321 


9.5973 


2.94645 


1.13122 


2777.2 


613754 


885 


783225 


693154125 


29.7489 


9.6010 


2.94694 


1.12994 


2780.3 


615143 


886 


784996 


695506456 


29.7658 


9.6046 


2.94743 


1.12867 


2783.5 


616534 


887 


786769 


697864103 


29.7825 


9.6082 


2.94792 


1.12740 


2786.6 


617927 


888 


788544 


700227072 


29.7993 


9.6118 


2.94841 


1.12613 


2789.7 


619321 


889 


790321 


702595369 


29.8161 


9.6154 


2.94890 


1.12486 


2792.9 


620717 


890 


792100 


704969000 


29.8329 


9.6190 


2.94939 


1.12360 


2796.0 


622114 


891 


793881 


707347971 


29.8496 


9.6226 


2.94988 


1.12233 


2799.2 


623513 


892 


795664 


709732288 


29.8664 


9.6262 


2.95036 


1.12108 


2802.3 


624913 


893 


797449 


712121957 


29.8831 


9.6298 


2.95085 


1.11982 


2805.4 


626315 


894 


799236 


714516984 


29.8998 


9.6334 


2.95134 


1.11857 


2808.6 


627718 


895 


801025 


716917375 


29.9166 


9.6370 


2.95182 


1.11732 


2811.7 


629124 


896 


802816 


719323136 


29.9333 


9.6406 


2.95231 


1.11607 


2814.9 


630530 


897 


804609 


721734273 


29.9500 


9.6442 


2.95279 


1.11483 


2818.0 


631938 


898 


806404 


724150792 


29.9666 


9.6477 


2.95328 


1.11359 


2821.2 


633348 


899 


808201 


726572699 


29.9833 


9.6513 


2.95376 


1.11235 


2824.3 


634760 



138 



SQUARES, CUBES, SQUARE ROOTS, CUBE ROOTS, LOGARITHMS, RECIPROCALS, 
CIRCUMFERENCES AND CIRCULAR AREAS OF NOS. FROM 1 TO 1000 



No. 


Square 


Cube 


Sq. Root 


Cu. Root 


Log. 


lOOOxRecip. 


No. = Dia. 


















Circuin. 


Area 


900 


810000 


729000000 


30.0000 


9.6549 


2.95424 


1.11111 


2827.4 


636173 


901 


811801 


731432701 


30.0167 


9.6585 


2.95472 


1.10988 


2830.6 


637587 


902 


813604 


733870808 


30.0333 


9.6620 


2.95521 


1.10865 


2833.7 


639003 


903 


815409 


736314327 


30.0500 


9.6656 


2.95569 


1.10742 


2830.9 


640421 


904 


817216 


738703264 


30.0666 


9.6692 


2.95617 


1.10619 


28-10.0 


641840 


905 


819025 


741217625 


30.0832 


9.6727 


2.95665 


1.10497 


2843.1 


643261 


906 


820836 


743677416 


30.0998 


9.6763 


2.95713 


1.10375 


2846.3 


644683 


907 


822649 


746142643 


30.1164 


9.6799 


2.95761 


1.10254 


2849.4 


646107 


908 


824464 


748613312 


3o.iaso 


9.6834 


2.95809 


1.10132 


2852.6 


647533 


909 


826281 


751089429 


30.1496 


9.6870 


2.95856 


1.10011 


2855.7 


648960 


910 


828100 


753571000 


30.1662 


9.6905 


2.95904 


1.09890 


2858.8 


6.50388 


911 


829921 


756058031 


30.1828 


9.6941 


2.95952 


1.09769 


2*62.0 


651818 


912 


&31744 


758550528 


30.1993 


9.6976 


2.95999 


1.09649 


2865.1 


653250 


913 


883569 


761048497 


30.2159 


9.7012 


2.96047 


1.09529 


2868.3 


654684 


914 


835396 


763551944 


30.2324 


9.7047 


2.96095 


1.09409 


2871.4 


656118 


915 


837225 


766060875 


30.2490 


9.7082 


2.96142 


1.09290 


2874.6 


657555 


916 


839056 


768575296 


30.2655 


9.7118 


2.96190 


1.09170 


2877.7 


658993 


917 


840889 


771095213 


30.2*20 


9.7153 


2.96237 


1.09051 


2880.8 


660433 


918 


842724 


773620632 


30.2985 


9.7188 


2.96284 


1.08932 


2884.0 


661874 


919 


844561 


776151559 


30.3150 


9.7224 


2.96332 


1.08814 


2887.1 


663317 


920 


846400 


778688000 


30.3315 


9.7259 


2.96379 


1.08696 


2890.3 


664761 


921 


848241 


781229961 


30.3480 


9.7294 


2.96426 


1.08578 


2893.4 


666207 


922 


850084 


783777448 


30.3645 


9.7329 


2.J6473 


1.08460 


2896.5 


667654 


923 


851929 


786330467 


30.3809 


9.7364 


2.96520 


1.08342 


2899.7 


669103 


924 


853776 


788889024 


30.3974 


9.7400 


2.96567 


1.08225 


2902.8 


670554 


925 


855625 


791453125 


30.4138 


9.7435 


2.96614 


1.08108 


2906.0 


672006 


926 


857476 


794022776 


30.4302 


9.7470 


2.96661 


1.07991 


2909.1 


673460 


927 


859329 


796597983 


30.4467 


9.7505 


2.96708 


1.07875 


2912.3 


674915 


928 


861184 


799178752 


30.4631 


9.7540 


2.96755 


1.07759 


2915.4 


676372 


929 


863041 


801765089 


30.4795 


9.7575 


2.96802 


1.07643 


2918.5 


677831 


930 


864900 


804357000 


30.4959 


9.7610 


2.96848 


1.07527 


2921.7 


679291 


931 


866761 


806954491 


30.5123 


9.7645 


2.96895 


1.07411 


2924.8 


680752 


932 


868624 


809557568 


30.5287 


9.7680 


2.96942 


1.07296 


2928.0 


682216 


933 


870489 


812166237 


30.5450 


9.7715 


2.96988 


1.07181 


2931.1 


683680 


934 


872356 


814780504 


30.5614 


9.7750 


2.97035 


1.07066 


2934.2 


685147 


935 


874225 


817400375 


30.5778 


9.7785 


2.97081 


1.06952 


2937.4 


686615 


936 


876096 


820025856 


30.5941 


9.7819 


2.97128 


1.06838 


2940.5 


688084 


937 


877969 


822656953 


30.6105 


9.7854 


2.97174 


1.06724 


2943.7 


689555 


938 


879844 


825293672 


30.6268 


9.7889 


2.97220 


1.06610 


2946.8 


691028 


939 


881721 


827936019 


30.6431 


9.7924 


2.97267 


1.06496 


2950.0 


692502 


940 


883600 


830584000 


30.6594 


9.7959 


2.97-313 


1.06383 


2953.1 


693978 


941 
942 
943 
944 
945 
946 
947 
948 
949 


885481 


833237621 


80.6757 


9.7993 


2.97359 


1.06270 


2956.2 


695455 


887864 


835896888 


30.6920 


9.8028 


2.97405 


1 .06157 


2959.4 


696934 


889249 
891136 
893025 
894916 
896809 
898704 
900601 


838561&D7 
841232384 


30.7083 
30.7246 


9.8053 
9.8097 


2.97451 
2.97497 


1.06045 
1.05932 


2962.5 
2965.7 


698415 
699897 


843908625 

846590536 
849278123 
851971392 
854670349 


30.7409 
30.7571 
30.7734 
30.7896 

30.8053 


9.8132 
9.8167 
9.8201 
9.8236 
9.8270 


2.97543 

2.97589 
2.97635 
2.97681 
2.97727 


1.05820 
1.05708 
1.05597 
1.05485 
1.05374 


2968.8 
2971.9 
2975.1 
2978.2 
2981.4 


701380 
702S65 
704352 
705840 
7073:30 



139 



SQUARES, CUBES, SQUARE ROOTS, CUBE ROOTS, LOGARITHMS, RECIPROCALS, 
CIRCUMFERENCES AND CIRCULAR AREAS OF NOS. FROM 1 TO 1000 



No. 


Square 


Cube 


Sq. Root 


Cu. Root 


Log. 


lOOOxRecip. 


No.= 


Dia. 








Circum. 


Area 


950 


902500 


857375000 


30.8221 


9.&305 


2.97772 


1.05263 


2984.5 


708822 


951 


904401 


860085351 


30.8383 


9.8339 


2.97818 


1.05152 


2987.7 


710315 


952 


906304 


862801408 


30.8545 


9.8374 


2.97864 


1.05042 


2990.8 


711809 


953 


908209 


865523177 


30.8707 


9.8408 


2.97909 


1.04932 


2993.9 


713306 


954 


910116 


868250664 


30.8869 


9.8443 


2.97955 


1.04822 


2997.1 


714803 


955 


912025 


870983875 


30.9031 


9.8477 


2.98000 


1.04712 


3000.2 


716303 


956 


913936 


873722816 


30.9192 


9.8511 


2.98046 


1.04603 


3003.4 


717804 


957 


915849 


876467493 


30.9354 


9.8546 


2.98091 


1.04493 


3006.5 


719306 


958 


917764 


879217912 


30.9516 


9.8580 


2.98137 


1.04384 


3009.6 


720810 


959 


919681 


881974079 


30.9677 


9.8614 


2.98182 


1.04275 


3012.8 


722316 


960 


921600 


884736000 


30.9839 


9.8648 


2.98227 


1.04167 


3015.9 


723823 


961 


923521 


887503681 


31.0000 


9.8683 


2.98272 


1.04058 


3019.1 


725332 


962 


925444 


890277128 


31.0161 


9.8717 


2.98318 


1.03950 


3022.2 


726842 


963 


927369 


893056347 


31.0322 


9.8751 


2.98363 


1.03842 


3025.4 


728354 


964 


929296 


895841344 


31.0483 


9.8785 


2.98408 


1.03734 


3028.5 


729867 


965 


931225 


898632125 


31.0644 


9.8819 


2.98453 


1.03627 


3031.6 


731382 


966 


933156 


901428696 


31.0805 


9.8854 


2.98498 


1.03520 


3034.8 


732899 


967 


935089 


904231063 


31.0966 


9.8888 


2.98543 


1.03413 


3037.9 


734417 


968 


937024 


907039232 


31.1127 


9.8922 


3.98588 


1.03306 


3041.1 


735937 


989 


938051 


909853209 


31.1288 


9.8956 


2.98632 


1.03199 


3044.2 


737458 


970 


940900 


912673000 


31.1448 


9.8990 


2.98677 


1.03093 


3047.3 


738981 


971 


942841 


915498611 


31.1609 


9.9024 


2.98722 


1.02987 


3050.5 


740506 


972 


944784 


918330048 


31.1769 


9.9058 


2.98767 


1.02881 


3053.6 


742032 


973 


946729 


921167317 


31.1929 


9.9092 


2.98811 


1.02775 


3056.8 


743559 


974 


948676 


924010424 


31.2090 


9.9126 


2.98858 


1.02669 


3059.9 


745088 


975 


950625 


926859375 


31.2250 


9.9160 


2.98900 


1.02564 


3063.1 


746619 


976 


952576 


929714176 


31.2410 


9.9194 


2.98945 


1.02459 


3066.2 


748151 


977 


954529 


932574833 


31.2570 


9.9227 


2.98989 


1.02354 


3069.3 


749685 


978 


956484 


935441352 


31.2730 


9.9261 


2.99034 


1.02249 


3072.5 


751221 


979 


958441 


938313739 


31.2890 


9.9295 


2.99078 


1.02145 


3075.6 


752758 


980 


960400 


941192000 


31.3050 


9.9329 


2.99123 


1.02041 


3078.8 


754296 


981 


962361 


944076141 


31.3209 


9.9363 


2.99167 


1.01937 


3081.9 


755837 


982 


964324 


946966168 


31.3369 


9.9396 


2.99211 


1.01833 


3085.0 


757378 


983 


966289 


949862087 


31.3528 


9.9430 


2.99255 


1.01729 


3088.2 


758922 


984 


968256 


952763904 


31.3688 


9.9464 


2.99300 


1.01626 


3091.3 


760466 


985 


970225 


955671625 


31.3847 


9.9497 


2.99344 


1.01523 


3094.5 


762013 


986 


972196 


958585256 


31.4006 


9.9531 


2.99388 


1.01420 


3097.6 


763561 


987 


974169 


961504803 


31.4166 


9.9565 


2.99432 


1.01317 


3100.8 


765111 


988 


976144 


964430272 


31.4325 


9.9598 


2.99476 


1.01215 


3103.9 


766662 


989 


978121 


967361669 


31.4484 


9.9632 


2.99520 


1.01112 


3107.0 


768214 


990 


980100 


970299000 


31.4643 


9.9666 


2.99564 


1.01010 


3110.2 


769769 


991 


982081 


973242271 


31.4802 


9.9699 


2.99607 


1.00908 


3113.3 


771325 


992 


984064 


976191488 


31.4960 


9.9733 


2.99651 


1.00806 


3116.5 


772882 


993 


986049 


979146657 


31.5119 


9.9766 


2.99695 


1.00705 


3119.6 


774441 


994 


988036 


982107784 


31.5278 


9.9800 


2.99739 


1.00604 


3122.7 


776002 


995 


990025 


985074875 


31.5436 


9.9833 


2.99782 


1.00503 


3125.9 


777504 


996 


992016 


988047936 


31.5595 


9.9866 


2.99826 


1.00402 


3129.0 


779128 


997 


994009 


991026973 


31.5753 


9.9900 


2.99870 


1.00301 


3132.2 


780693 


998 


996004 


994011992 


31.5911 


9.9933 


2.99913 


1.00200 


3135.3 


782260 


999 


998001 


997002999 


31.6070 


9.9967 


2.99957 


1.00100 


3138.5 


783828 



141 



STRUCTURES 

The photographs following show some of the 
structures wherein du Mazuel reenforcements have 
been used to advantage and in which, in some in- 
stances, they were the only materials that could be 
utilized . 




n 



NEW YORK COTTON EXCHANGE 
William and Beaver Streets, Borough of Manhattan 

Qf ?°°j ?/ addition for cotton sampling. Made of 3 inch slabs with 
Mandard No. 26 du Mazuel reenforcement, 13 foot spans; du Mazuel 
bneets overlapped at center of spans 

George Nichols, Architect S. E. McDonald, Contractor 



143 




BRETTON HALL 
Broadway and Eighty-sixth Street, Borough of Manhattan 
New York City 
Roof of new kitchens, 2 l / 2 inch slabs with Standard No. 26 du 
Mazuel reenforcement. 

S. E. McDonald, Contractor 



144 




INTERBOROUGH RAPID TRANSIT COMPANY 

George H. Pegram, Chief Engineer 

Subway Division 

Bowling Green Station, Borough of Manhattan, New York City 

Extension of the station necessitated taking advantage of every 
inch of space and getting strongest possible construction; the Stan- 
dard No. 26 du Mazuel reenforcement was therefore used in the 
storerooms 



145 




Storerooms Nos. 1 and 2 during construction 



146 





*i 



End storeroom and decorator's laboratory 




INTERBOROUGH RAPID TRANSIT COMPANY 

George H. Pegram, Chief Engineer 

Elevated Division 

Employees' Recreation Building, Fordham, Borough of Bronx, 
New York City. 




A happy lot of Interborough employees enjoying a quiet hour" 

under the protection of a 2/ 2 inch du Mazuel roof in the 

Fordham Recreation Building 




INTERBOROUGH RAPID TRANSIT COMPANY 
George H. Pegram, Chief Engineer 

One Hundred and Twenty-ninth Street and Third Avenue Inspec- 
tion Sheds and Club Rooms, Borough of Manhattan, New York City. 



150 




FRONT INTERIOR VIEW OF INSPECTION SHEDS 



Floors and pits made of 2 and 3 inch slabs with Standard No. 26 
du Mazuel reenforcement. An interesting experiment conducted in 
this shop showed that concrete cannot be used for manhole covers 



151 




REAR INTERIOR VIEW OF INSPECTION SHEDS 



A study of this photograph will give an idea of the extremely 
heavy service to which the 2 and 3 inch du Mazuel slabs are subjected 




ST. EDMUND'S CHURCH OF NEW YORK CITY 
Borough of Bronx 

Rev. Dr. J. C. Smiley, Rector 

Dr. E. G. F. R — du Mazuel, Engineer and Architect 

This church is erected entirely in the du Mazuel system of reen- 
forced concrete construction. Building not quite finished when 
Manual went to press. The complete cost of this building is less 
than $2r,. ooo 




153 



154 




RESIDENCE AND STUDIO OF MR. AND MISS STONE 

Stapleton, Staten Island 

Dr. E. G. F. R — du Mazuel, Engineer and Architect 

This building is entirely of du Mazuel reenforced concrete con- 
struction. All walls are two inches in thickness. 

It stands 300 feet above the sea level and is subjected at all times 
to great wind stresses. 



156 






V 




ipii- 



A corner of the boudoir on second floor. 



156 




The hall and staircase from the studio 



INDEX 



159 



Page 



Abattoirs 14 

Abutments 16, 72, 76 

Advantages of Reenf orced Concrete 6 

Alexandria 6 

Alkalies 11, 28 

Approaches for Arches 75 

Arches 72 

Arch Piers 73 

Arch Springs 76 

Arch Table 78 

Area of Circles 120 

Armories . 14 

Asylums 14 



B 



Barges .14, 16 

Barns 14 

Barracks 14 

Basic Color 9 

Bath Houses 14 

Beams 24 

Bearing Power of Earths 70, 71 

Bending Moments 54, 57 

Boards 24 

Boilers 16 

Bowling Green Station 144, 145, 146 

Bretton Hall 143 

Breweries 14 

Brick Building 5 

Bridge, 150 Foot du Mazuel Highway 74, 77 

Bridge Work 16, 27, 58 

Building Materials 5 

Bungalows 14 

Burnt Lime 29 

Buttresses 16 



Calcium Chloride 12 

Calculations for Concrete Members 51 

Canal Boats 16 

Carnegie Coefficients 59, 60, 61 

Carnegie Standards 59, 60, 61 



160 



Page 

Ceilings 16, 25, 26, 28, 34, 56 

Cement 6 

Centering of Wooden Beams 49 

Chair Rails 34 

Chimneys 16 

Churches 14 

Cinder Concrete 8, 28 

Cinders 28 

Circumferences 120 

Cisterns 11 

Coal Bins 16 

Coffer Dams 16 

Collar Beams 48 

Collapse of Slabs 29 

Colleges 14 

Coloring of Concrete and Mortar 9 

Columns 24, 55 

Comparative Costs 13,14, 15 

Common Radius 17 

Compression in Slabs 28, 29 

Concrete 5, 28 

Concrete Coloring 9 

Concrete, du Mazuel Oleaginous 11, 76 

Concrete Formulae 28, 29, 52, 53 

Concrete Freezing 12, 33 

Concrete Interlocking Steel Corporation 1 

Concrete Members 51 

Concrete Mixtures 8, 9, 10 

Concrete, Plain and Reenforced 5 

Concrete Setting 11 

Concrete, Sustaining Power of 47 

Concrete Tensile Strength 11 

Concrete, Value of 5 

Concrete Waterproofing 11 

Conduits 11 

Corrosion 28, 29, 33 

Cosecants 119 

Cosines 115 

Cost Reduction in a Building 42 

Costs of Buildings 13, 14, 15 

Cotangents 117 

Cottages 14 

Crown of an Arch 72, 77 

Crushed Stone 8 

Cubes 120 

Cube Roots 120 

Culverts . . .- 16 



161 



Page 

Dams 14, 16, 24, 81 

Decomposition of Building Materials 6 

Decimals of a Foot 108 

Decimals of an Inch 107 

Deflections 54, 55 

Dehydration 29 

Delta 55 

Design of Arches 72 

Diagrams for Slabs 43, 44, 45, 46 

Door Jams and Frames 34 

Drinking Troughs 11 

Du Mazuel Arch 73 

Du Mazuel No. 22 Slabs 43 

Du Mazuel No. 24 Slabs 44 

Du Mazuel No. 26 Slabs 45 

Du Mazuel No. 28 Slabs 46 

Du Mazuel Oleaginous Concrete 11, 76 

Du Mazuel Reenforcements (Common Radius) 17 

Du Mazuel Reenforcements (Gauges) 17 

Du Mazuel Reenforcements (Tests) 8, 29, 84 

Du Mazuel Reenforcements 7, 16, 18, 19, 20, 21 

Dwellings 14 



Egyptian Cements 6 

Emulsion of Oil and Alkalies 11 

Example of du Mazuel Arch 73 

Expansion in Arches 76 



Factories 14, 58 

Fastening to Supporting Structure 24 

Feeding Floors 11 

Felt, When Needed 29 

Fences 36 

Final Setting 11 

Fire Tests 29 

Fire and Load Tests 84 

Fire Places 16 

Flashing 33 

Flat Roofs 33 

Flats 14 

Floor Before Concreting 24 

Floor Example 62 

Floor Finish (Loads) 40 



F 

Page 

Floors 16, 24, 25, 26, 27, 40 

Flues 16 

Fordham Recreation Building 147, 148 

Forms 17, 24 

Formulae for Beams 54 

Formulae for Cement of Ancients 6 

Formulae for Concrete 52, 53 

Form Work 6, 7, 24 

Fortifications 14 

Foundations 70, 73 

Frame Buildings : 5, 14, 15 

Frame Work 24 

Free Lime 28 

Freezing 12, 33 

Frost 12, 33 

Full Bearing Capacity of Slabs 47 

Furring ; 37 

Furring Strips 24 

G 

Galvanizing 28 

Garages 14 

H 

Hair 29 

Haunches of an Arch 76 

Heating Systems 82, 83 

Hoppers 16 

Horizontal Uniform Pressure in Arch 79 

Hospitals 15 

Hotels 15 

Hot Houses 15 

Hydraulic Cement 28 

I 

Ichnographic Section of an Arch 75 

Initial Setting 11 

Inspection of Work 3 

Inspection Shed 149, 150, 151 

Interborough Rapid Transit Co 144, 147, 149 

Interlocking 22, 29 

Intrados 76 

Iron Ore Tailings 8 

Iron Oxide 9 

Irons, Hot, for Waterproofing 33 

Is a Floor Before Concreting 24 



163 



Page 

Joints 29 

Joining Sheets 22 

K 

Keystone 72 



Libraries 15 

Lime 28, 29 

Line of Pressure of an Arch 72, 79 

Loading (Standard) 40 

Logarithms by Dr. du Mazuel 102 

Logarithms to Five Spaces • 120 

Log Buildings 5 

Low Cost 16 

M 

Main Beams 63 

Manufacturers of du Mazuel Reenforcements 1 

Manure Wells or Sinks 11 

Maximum Fibre Strain of Wooden Beams 48 

Maximum Loads 54 

Maximum Strength, When reached 47 

Metallurgical Hair 29 

Mills 15, 58 

Mineral Oil 11 

Mixtures 8, .9, 10, 29 

Molecular Moisture 29 

Mongolian Cements 6 

Mortars 8, 9, 10 

Moldings for Bridges 77 

Municipal Buildings 15 

Museums 15 

N 

New York Cotton Exchange 142 

Non-volatile Mineral Oil 11 

Notes on the Design of Floors 40 

Nummulites 5 



Office Buildings 58 

Oil 11 

Oleaginous du Mazuel Concrete 11, 76 

Overlapping .' 22 



Page 

Painting 28 

Paraffin Wax 33 

Partitions 16, 24, 33, 34, 35 

Partitions, Reenforcement placed Horizontally 33 

Penstock 16 

Percolating Waters 28 

Permits 93 

Physical Properties of Metals 100 

Picture Mouldings 34 

Piers for Arches 73 

Pipe 32 

Pitch Roofs 24, 29, 30, 31 

Pit Lining and Sheeting 17 

Plan of a Factory Floor 64, 65 

Planks 24 

Plastered Ceilings 56 

Plastering 28, 29, 33, 34, 41 

Pores 33 

Porous Materials 28 

Portland Cement 28 

Power Houses 15 

Preface 3 

Pressure on Foundations 71 

Prisons 15 

Proportions 9 

Protection for Steel 28 

Public Buildings 15 

Purlins 24 

Putrefaction 11 

Pyramids of Egypt 5 



Railings 77 

Railway Stations 15 

Reactions 56 

Reciprocals 120 

Reduction of Steel Work in a Building 42 

Reenforced Concrete in America 3 

Reenf orcements 6 

Religious Rites of the Egyptians 5 

Reservoirs 15 

Residences 58 

Retaining Walls 16, 24 

Ring of an Arch 72 

Roadbed 77 

Rolling Loads 58 



165 

R 

Page 

Roman Cement 6 

Roofing 29 

Roofs 11, 16, 24, 29, 30, 31, 33 

Rule for Selecting Beams 63 

Runways 24 



Safe Live Loads, Dead Loads, Weights, etc 42 

Safe Live Loads, Dead Loads, Weights, etc. (Curves). 43 

Safe Working Factors 58 

Secants 118 

Secondary Beams 63 

Section Moduli 53, 54, 58, 60, 61 

Selection of Materials 8 

Self Interlocking 16 

Serpentine Rocks 5 

Setting of Concrete 11 

Sewers 11, 15, 24, 32 

Shear 57 

Sheet Piling 16, 17, 80 

Shelter for Materials 16 

Sidewalk Slabs 16 

Siding 34 

Silos 11 

Sines, Cosines Tangents, etc 114 

Sinks for Manure 11 

Size of Wooden Beams for Various Loads 49 

Skewbacks 77 

Skilled Labor 16 

Skylights 33 

Skyscrapers 15 

Slabs 62 

Sliding . . 24 

Spacing of Structural Members 41 

Spacing of Wooden Beams 49, 50 

Spandrel Filling 77 

Spreading Base 70 

Square Roots 120 

Squares 120 

St. Edmund's Church of New York City 152, 153 

Stables 15 

Stack Openings 33 

Standard Gauges for Sheet Metal 99 

Standard Loading 40 

Stone Buildings 5 

Stone's Residence, Mr. and Miss 154, 155, 156 



Page 

Stores 15 

Storing Reenf orcements 28 

Structures 141 

Sun, Effect of 33 

Supports 24 

Sustaining Power of Concrete 47 

Sustaining Power of Soils TO, 71 



Tangents 116 

Tapering Base for Foundations 70 

Temperature 12 

Temples 15 

Tensile Strength in Concrete 11 

Tension Zone 29 

Terrazzo Floors 11 

Tests of du Mazuel Reenforcements 8, 29, 84 

Theatres 15 

Thickness of Walls, non du Mazuel 38, 39 

Trench Lining and Sheeting 17 

Triangle Formulae 112, 113 

Tunnel Tubing 17 

Typical Example of a Slab 62 

U 

Unit Strains on Wooden Beams 49 

Use Without Forms 17 



Value of Concrete 5 

Ventilator Openings 33 

Voids in Concrete 8 

Vousoirs 72 

W 

Walls 16, 17, 24, 35, 36, 37, 38, 39 

Walls, Reenforcement Placed Horizontally 33 

Warehouses 15 

Waste Product 8 

Water 11 

Waterproofing 29 

Watertight Surfaces 29 

Wax 33 



w 

Page 

Weight of Substances 96 

Wells for Manure 11 

Wheelbarrows 24 

White Oil 11 

Winter Concrete 12, 33 

Wooden Beams 48 

Workmen 24 

Worm Cracks 11 



NOTES 



NOTES 



NOTES 



NOTES 



NOTES 



NOTES 



NOTES 



NOTES 



NOTES 



DEC 17 J9J0 



One copy del. to Cat. Div. 
1 ' f? 1910 



